Distance on a great circle

Percentage Accurate: 62.3% → 98.7%

Time: 36.2s

Alternatives: 32

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0))))
   (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	return R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0)
    code = r * (2.0d0 * atan2(sqrt(t_1), sqrt((1.0d0 - t_1))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_1 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * t_0) * t_0);
	return R * (2.0 * Math.atan2(Math.sqrt(t_1), Math.sqrt((1.0 - t_1))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0)
	return R * (2.0 * math.atan2(math.sqrt(t_1), math.sqrt((1.0 - t_1))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = (sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 62.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0))))
   (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	return R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0)
    code = r * (2.0d0 * atan2(sqrt(t_1), sqrt((1.0d0 - t_1))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_1 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * t_0) * t_0);
	return R * (2.0 * Math.atan2(Math.sqrt(t_1), Math.sqrt((1.0 - t_1))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0)
	return R * (2.0 * math.atan2(math.sqrt(t_1), math.sqrt((1.0 - t_1))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = (sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right)\\ t_1 := \sin \left(-0.5 \cdot \lambda_2\right) \cdot \cos \left(\lambda_1 \cdot 0.5\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left({t\_0}^{2} - -2 \cdot \left(t\_0 \cdot t\_1\right)\right) + {t\_1}^{2}\right), {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (* lambda1 0.5)) (cos (* lambda2 0.5))))
        (t_1 (* (sin (* -0.5 lambda2)) (cos (* lambda1 0.5))))
        (t_2
         (fma
          (cos phi1)
          (*
           (cos phi2)
           (+ (- (pow t_0 2.0) (* -2.0 (* t_0 t_1))) (pow t_1 2.0)))
          (pow
           (-
            (* (cos (* 0.5 phi2)) (sin (* 0.5 phi1)))
            (* (cos (* 0.5 phi1)) (sin (* 0.5 phi2))))
           2.0))))
   (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 * 0.5)) * cos((lambda2 * 0.5));
	double t_1 = sin((-0.5 * lambda2)) * cos((lambda1 * 0.5));
	double t_2 = fma(cos(phi1), (cos(phi2) * ((pow(t_0, 2.0) - (-2.0 * (t_0 * t_1))) + pow(t_1, 2.0))), pow(((cos((0.5 * phi2)) * sin((0.5 * phi1))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))), 2.0));
	return R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(Float64(lambda1 * 0.5)) * cos(Float64(lambda2 * 0.5)))
	t_1 = Float64(sin(Float64(-0.5 * lambda2)) * cos(Float64(lambda1 * 0.5)))
	t_2 = fma(cos(phi1), Float64(cos(phi2) * Float64(Float64((t_0 ^ 2.0) - Float64(-2.0 * Float64(t_0 * t_1))) + (t_1 ^ 2.0))), (Float64(Float64(cos(Float64(0.5 * phi2)) * sin(Float64(0.5 * phi1))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(lambda2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[(-2.0 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 62.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 61.4%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
4. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Applied rewrites 62.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
6. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

7. Applied rewrites 62.5%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
8. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

9. Applied rewrites 77.3%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
10. Step-by-step derivation

    1. lift-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    2. lift-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    3. lift--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    4. div-sub N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    5. sin-diff N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    6. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    7. mult-flip-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    8. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    9. *-commutative N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    10. lift-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    11. lift-sin.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    12. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    13. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    14. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    15. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    16. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    17. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    18. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    19. mult-flip-rev N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    20. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    21. *-commutative N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    22. lift-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    23. lower-sin.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    24. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    25. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

11. Applied rewrites 78.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
12. Step-by-step derivation

    1. lift-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    2. lift-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    3. lift--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    4. div-sub N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    5. sin-diff N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    6. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    7. mult-flip-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    8. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    9. *-commutative N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    10. lift-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    11. lift-sin.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    12. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    13. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    14. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    15. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    16. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    17. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    18. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    19. mult-flip-rev N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    20. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    21. *-commutative N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    22. lift-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    23. lower-sin.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    24. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    25. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

13. Applied rewrites 98.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

14. Taylor expanded in lambda1 around inf

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
15. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

16. Applied rewrites 98.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

17. Taylor expanded in lambda1 around inf

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
18. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

19. Applied rewrites 98.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

21. Applied rewrites 98.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left({\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right)\right)}^{2} - -2 \cdot \left(\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right)\right) \cdot \left(\sin \left(-0.5 \cdot \lambda_2\right) \cdot \cos \left(\lambda_1 \cdot 0.5\right)\right)\right)\right) + \color{blue}{{\left(\sin \left(-0.5 \cdot \lambda_2\right) \cdot \cos \left(\lambda_1 \cdot 0.5\right)\right)}^{2}}\right), {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

23. Applied rewrites 98.6%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left({\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right)\right)}^{2} - -2 \cdot \left(\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right)\right) \cdot \left(\sin \left(-0.5 \cdot \lambda_2\right) \cdot \cos \left(\lambda_1 \cdot 0.5\right)\right)\right)\right) + {\left(\sin \left(-0.5 \cdot \lambda_2\right) \cdot \cos \left(\lambda_1 \cdot 0.5\right)\right)}^{2}\right), {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left({\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right)\right)}^{2} - -2 \cdot \left(\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right)\right) \cdot \left(\sin \left(-0.5 \cdot \lambda_2\right) \cdot \cos \left(\lambda_1 \cdot 0.5\right)\right)\right)\right) + \color{blue}{{\left(\sin \left(-0.5 \cdot \lambda_2\right) \cdot \cos \left(\lambda_1 \cdot 0.5\right)\right)}^{2}}\right), {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
24. Add Preprocessing

Alternative 2: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (*
           (cos phi2)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
            2.0))
          (pow
           (-
            (* (cos (* 0.5 phi2)) (sin (* 0.5 phi1)))
            (* (cos (* 0.5 phi1)) (sin (* 0.5 phi2))))
           2.0))))
   (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi1), (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0)), pow(((cos((0.5 * phi2)) * sin((0.5 * phi1))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))), 2.0));
	return R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi1), Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0)), (Float64(Float64(cos(Float64(0.5 * phi2)) * sin(Float64(0.5 * phi1))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 61.4%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
4. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Applied rewrites 62.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
6. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

7. Applied rewrites 62.5%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
8. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

9. Applied rewrites 77.3%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
10. Step-by-step derivation

    1. lift-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    2. lift-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    3. lift--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    4. div-sub N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    5. sin-diff N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    6. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    7. mult-flip-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    8. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    9. *-commutative N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    10. lift-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    11. lift-sin.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    12. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    13. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    14. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    15. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    16. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    17. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    18. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    19. mult-flip-rev N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    20. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    21. *-commutative N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    22. lift-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    23. lower-sin.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    24. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    25. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

11. Applied rewrites 78.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
12. Step-by-step derivation

    1. lift-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    2. lift-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    3. lift--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    4. div-sub N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    5. sin-diff N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    6. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    7. mult-flip-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    8. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    9. *-commutative N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    10. lift-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    11. lift-sin.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    12. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    13. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    14. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    15. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    16. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    17. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    18. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    19. mult-flip-rev N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    20. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    21. *-commutative N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    22. lift-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    23. lower-sin.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    24. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    25. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

13. Applied rewrites 98.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

14. Taylor expanded in lambda1 around inf

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
15. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

16. Applied rewrites 98.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

17. Taylor expanded in lambda1 around inf

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
18. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

19. Applied rewrites 98.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
20. Add Preprocessing

Alternative 3: 88.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \lambda_1\right)\\ t_1 := \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), t\_0, \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_2 := \mathsf{fma}\left(\cos \phi_1, t\_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ t_4 := {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ \mathbf{if}\;\lambda_2 \leq -7.8 \cdot 10^{-16}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\lambda_2 \leq 0.003:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_1, t\_4\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {t\_0}^{2}, t\_4\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 lambda1)))
        (t_1
         (*
          (cos phi2)
          (pow
           (fma
            (cos (* -0.5 lambda2))
            t_0
            (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
           2.0)))
        (t_2
         (fma
          (cos phi1)
          t_1
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5)))))))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2))))))
        (t_4
         (pow
          (-
           (* (cos (* 0.5 phi2)) (sin (* 0.5 phi1)))
           (* (cos (* 0.5 phi1)) (sin (* 0.5 phi2))))
          2.0)))
   (if (<= lambda2 -7.8e-16)
     t_3
     (if (<= lambda2 0.003)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (fma (cos phi1) t_1 t_4))
          (sqrt (- 1.0 (fma (cos phi1) (* (cos phi2) (pow t_0 2.0)) t_4))))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * lambda1));
	double t_1 = cos(phi2) * pow(fma(cos((-0.5 * lambda2)), t_0, (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0);
	double t_2 = fma(cos(phi1), t_1, (0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))))));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double t_4 = pow(((cos((0.5 * phi2)) * sin((0.5 * phi1))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))), 2.0);
	double tmp;
	if (lambda2 <= -7.8e-16) {
		tmp = t_3;
	} else if (lambda2 <= 0.003) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), t_1, t_4)), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * pow(t_0, 2.0)), t_4)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * lambda1))
	t_1 = Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), t_0, Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0))
	t_2 = fma(cos(phi1), t_1, Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5))))))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	t_4 = Float64(Float64(cos(Float64(0.5 * phi2)) * sin(Float64(0.5 * phi1))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0
	tmp = 0.0
	if (lambda2 <= -7.8e-16)
		tmp = t_3;
	elseif (lambda2 <= 0.003)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), t_1, t_4)), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * (t_0 ^ 2.0)), t_4))))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * t$95$0 + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * t$95$1 + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[lambda2, -7.8e-16], t$95$3, If[LessEqual[lambda2, 0.003], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$1 + t$95$4), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -7.79999999999999954e-16 or 0.0030000000000000001 < lambda2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 75.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 74.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}\right) \]

   if -7.79999999999999954e-16 < lambda2 < 0.0030000000000000001

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

   20. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   21. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   22. Applied rewrites 60.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 87.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\\ t_1 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2\\ \mathbf{if}\;\lambda_2 \leq -7.8 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\lambda_2 \leq 0.003:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (*
           (cos phi2)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
            2.0))
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5)))))))
        (t_1 (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))))))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3
         (+
          (pow
           (-
            (* (sin (* phi1 0.5)) (cos (* phi2 0.5)))
            (* (cos (* phi1 0.5)) (sin (* phi2 0.5))))
           2.0)
          (* (* (* (cos phi1) (cos phi2)) t_2) t_2))))
   (if (<= lambda2 -7.8e-16)
     t_1
     (if (<= lambda2 0.003)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi1), (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0)), (0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))))));
	double t_1 = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = pow(((sin((phi1 * 0.5)) * cos((phi2 * 0.5))) - (cos((phi1 * 0.5)) * sin((phi2 * 0.5)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_2) * t_2);
	double tmp;
	if (lambda2 <= -7.8e-16) {
		tmp = t_1;
	} else if (lambda2 <= 0.003) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi1), Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5))))))
	t_1 = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64((Float64(Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(phi2 * 0.5))) - Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 * 0.5)))) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2))
	tmp = 0.0
	if (lambda2 <= -7.8e-16)
		tmp = t_1;
	elseif (lambda2 <= 0.003)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -7.8e-16], t$95$1, If[LessEqual[lambda2, 0.003], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -7.79999999999999954e-16 or 0.0030000000000000001 < lambda2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 75.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 74.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}\right) \]

   if -7.79999999999999954e-16 < lambda2 < 0.0030000000000000001

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 63.2%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 78.7%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 87.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\lambda_2 \leq -7.8 \cdot 10^{-16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_2 \leq 0.003:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (* (cos phi2) (/ 1.0 (pow (sin (* (- lambda2 lambda1) -0.5)) -2.0)))
          (pow
           (-
            (* (cos (* 0.5 phi2)) (sin (* 0.5 phi1)))
            (* (cos (* 0.5 phi1)) (sin (* 0.5 phi2))))
           2.0)))
        (t_1
         (fma
          (cos phi1)
          (*
           (cos phi2)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
            2.0))
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5)))))))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= lambda2 -7.8e-16)
     t_2
     (if (<= lambda2 0.003)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi1), (cos(phi2) * (1.0 / pow(sin(((lambda2 - lambda1) * -0.5)), -2.0))), pow(((cos((0.5 * phi2)) * sin((0.5 * phi1))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))), 2.0));
	double t_1 = fma(cos(phi1), (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0)), (0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))))));
	double t_2 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double tmp;
	if (lambda2 <= -7.8e-16) {
		tmp = t_2;
	} else if (lambda2 <= 0.003) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi1), Float64(cos(phi2) * Float64(1.0 / (sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ -2.0))), (Float64(Float64(cos(Float64(0.5 * phi2)) * sin(Float64(0.5 * phi1))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0))
	t_1 = fma(cos(phi1), Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5))))))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	tmp = 0.0
	if (lambda2 <= -7.8e-16)
		tmp = t_2;
	elseif (lambda2 <= 0.003)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(1.0 / N[Power[N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -7.8e-16], t$95$2, If[LessEqual[lambda2, 0.003], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -7.79999999999999954e-16 or 0.0030000000000000001 < lambda2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 75.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 74.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}\right) \]

   if -7.79999999999999954e-16 < lambda2 < 0.0030000000000000001

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 79.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{\color{blue}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)}^{-2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 78.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{\color{blue}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)\\ t_1 := \cos \phi_2 \cdot \cos \phi_1\\ t_2 := t\_1 + \sin \phi_2 \cdot \sin \phi_1\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_4 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\\ \mathbf{if}\;\phi_1 \leq -0.25:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_4, t\_1, \mathsf{fma}\left(t\_2, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - t\_2 \cdot -0.5\right) - t\_4 \cdot t\_1}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 9.8 \cdot 10^{-82}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (*
           (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda2 lambda1) -0.5)))))
           (cos phi2))
          (pow
           (-
            (* (cos (* 0.5 phi2)) (sin (* 0.5 phi1)))
            (* (cos (* 0.5 phi1)) (sin (* 0.5 phi2))))
           2.0)))
        (t_1 (* (cos phi2) (cos phi1)))
        (t_2 (+ t_1 (* (sin phi2) (sin phi1))))
        (t_3
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (*
           (cos phi2)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
            2.0))))
        (t_4 (fma -0.5 (cos (- lambda2 lambda1)) 0.5)))
   (if (<= phi1 -0.25)
     (*
      (*
       (atan2
        (sqrt (fma t_4 t_1 (fma t_2 -0.5 0.5)))
        (sqrt (- (- 0.5 (* t_2 -0.5)) (* t_4 t_1))))
       2.0)
      R)
     (if (<= phi1 9.8e-82)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi1), ((0.5 - (0.5 * cos((2.0 * ((lambda2 - lambda1) * -0.5))))) * cos(phi2)), pow(((cos((0.5 * phi2)) * sin((0.5 * phi1))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))), 2.0));
	double t_1 = cos(phi2) * cos(phi1);
	double t_2 = t_1 + (sin(phi2) * sin(phi1));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0));
	double t_4 = fma(-0.5, cos((lambda2 - lambda1)), 0.5);
	double tmp;
	if (phi1 <= -0.25) {
		tmp = (atan2(sqrt(fma(t_4, t_1, fma(t_2, -0.5, 0.5))), sqrt(((0.5 - (t_2 * -0.5)) - (t_4 * t_1)))) * 2.0) * R;
	} else if (phi1 <= 9.8e-82) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi1), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda2 - lambda1) * -0.5))))) * cos(phi2)), (Float64(Float64(cos(Float64(0.5 * phi2)) * sin(Float64(0.5 * phi1))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0))
	t_1 = Float64(cos(phi2) * cos(phi1))
	t_2 = Float64(t_1 + Float64(sin(phi2) * sin(phi1)))
	t_3 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0)))
	t_4 = fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)
	tmp = 0.0
	if (phi1 <= -0.25)
		tmp = Float64(Float64(atan(sqrt(fma(t_4, t_1, fma(t_2, -0.5, 0.5))), sqrt(Float64(Float64(0.5 - Float64(t_2 * -0.5)) - Float64(t_4 * t_1)))) * 2.0) * R);
	elseif (phi1 <= 9.8e-82)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[phi1, -0.25], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$4 * t$95$1 + N[(t$95$2 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(t$95$2 * -0.5), $MachinePrecision]), $MachinePrecision] - N[(t$95$4 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 9.8e-82], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.25

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   11. Applied rewrites 57.1%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \left(\phi_2 - \phi_1\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \color{blue}{\left(\phi_2 - \phi_1\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       6. *-commutative N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       7. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       8. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       9. lower-+.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       10. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       11. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       12. *-commutative N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       14. lower-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2 \cdot \sin \phi_1}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       15. lower-sin.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2} \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       16. lower-sin.f64 58.1%

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \color{blue}{\sin \phi_1}, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   13. Applied rewrites 58.1%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]
   14. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       6. *-commutative N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       7. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       8. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       9. lower-+.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       10. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       11. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       12. *-commutative N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       14. lower-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       15. lower-sin.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       16. lower-sin.f64 73.6%

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   15. Applied rewrites 73.6%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   if -0.25 < phi1 < 9.8000000000000006e-82

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 63.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   14. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

   15. Applied rewrites 60.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}\right)}}\right) \]

   if 9.8000000000000006e-82 < phi1

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 76.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \color{blue}{\cos \phi_2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 76.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \color{blue}{\cos \phi_2}, {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 87.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := t\_0 + \sin \phi_2 \cdot \sin \phi_1\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_3 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\\ t_4 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_3, t\_0, \mathsf{fma}\left(t\_1, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - t\_1 \cdot -0.5\right) - t\_3 \cdot t\_0}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -0.25:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\phi_1 \leq 5.4 \cdot 10^{-36}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (+ t_0 (* (sin phi2) (sin phi1))))
        (t_2
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (*
           (cos phi2)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
            2.0))))
        (t_3 (fma -0.5 (cos (- lambda2 lambda1)) 0.5))
        (t_4
         (*
          (*
           (atan2
            (sqrt (fma t_3 t_0 (fma t_1 -0.5 0.5)))
            (sqrt (- (- 0.5 (* t_1 -0.5)) (* t_3 t_0))))
           2.0)
          R)))
   (if (<= phi1 -0.25)
     t_4
     (if (<= phi1 5.4e-36)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = t_0 + (sin(phi2) * sin(phi1));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0));
	double t_3 = fma(-0.5, cos((lambda2 - lambda1)), 0.5);
	double t_4 = (atan2(sqrt(fma(t_3, t_0, fma(t_1, -0.5, 0.5))), sqrt(((0.5 - (t_1 * -0.5)) - (t_3 * t_0)))) * 2.0) * R;
	double tmp;
	if (phi1 <= -0.25) {
		tmp = t_4;
	} else if (phi1 <= 5.4e-36) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = Float64(t_0 + Float64(sin(phi2) * sin(phi1)))
	t_2 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0)))
	t_3 = fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)
	t_4 = Float64(Float64(atan(sqrt(fma(t_3, t_0, fma(t_1, -0.5, 0.5))), sqrt(Float64(Float64(0.5 - Float64(t_1 * -0.5)) - Float64(t_3 * t_0)))) * 2.0) * R)
	tmp = 0.0
	if (phi1 <= -0.25)
		tmp = t_4;
	elseif (phi1 <= 5.4e-36)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[ArcTan[N[Sqrt[N[(t$95$3 * t$95$0 + N[(t$95$1 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -0.25], t$95$4, If[LessEqual[phi1, 5.4e-36], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.25 or 5.40000000000000015e-36 < phi1

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   11. Applied rewrites 57.1%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \left(\phi_2 - \phi_1\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \color{blue}{\left(\phi_2 - \phi_1\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       6. *-commutative N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       7. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       8. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       9. lower-+.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       10. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       11. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       12. *-commutative N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       14. lower-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2 \cdot \sin \phi_1}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       15. lower-sin.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2} \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       16. lower-sin.f64 58.1%

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \color{blue}{\sin \phi_1}, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   13. Applied rewrites 58.1%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]
   14. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       6. *-commutative N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       7. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       8. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       9. lower-+.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       10. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       11. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       12. *-commutative N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       14. lower-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       15. lower-sin.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       16. lower-sin.f64 73.6%

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   15. Applied rewrites 73.6%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   if -0.25 < phi1 < 5.40000000000000015e-36

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 63.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   14. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

   15. Applied rewrites 60.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := t\_0 + \sin \phi_2 \cdot \sin \phi_1\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_3 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\\ t_4 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_3, t\_0, \mathsf{fma}\left(t\_1, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - t\_1 \cdot -0.5\right) - t\_3 \cdot t\_0}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -1.05 \cdot 10^{-17}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\phi_2 \leq 10400000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (+ t_0 (* (sin phi2) (sin phi1))))
        (t_2
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (*
           (cos phi1)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
            2.0))))
        (t_3 (fma -0.5 (cos (- lambda2 lambda1)) 0.5))
        (t_4
         (*
          (*
           (atan2
            (sqrt (fma t_3 t_0 (fma t_1 -0.5 0.5)))
            (sqrt (- (- 0.5 (* t_1 -0.5)) (* t_3 t_0))))
           2.0)
          R)))
   (if (<= phi2 -1.05e-17)
     t_4
     (if (<= phi2 10400000.0)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = t_0 + (sin(phi2) * sin(phi1));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0));
	double t_3 = fma(-0.5, cos((lambda2 - lambda1)), 0.5);
	double t_4 = (atan2(sqrt(fma(t_3, t_0, fma(t_1, -0.5, 0.5))), sqrt(((0.5 - (t_1 * -0.5)) - (t_3 * t_0)))) * 2.0) * R;
	double tmp;
	if (phi2 <= -1.05e-17) {
		tmp = t_4;
	} else if (phi2 <= 10400000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = Float64(t_0 + Float64(sin(phi2) * sin(phi1)))
	t_2 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0)))
	t_3 = fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)
	t_4 = Float64(Float64(atan(sqrt(fma(t_3, t_0, fma(t_1, -0.5, 0.5))), sqrt(Float64(Float64(0.5 - Float64(t_1 * -0.5)) - Float64(t_3 * t_0)))) * 2.0) * R)
	tmp = 0.0
	if (phi2 <= -1.05e-17)
		tmp = t_4;
	elseif (phi2 <= 10400000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[ArcTan[N[Sqrt[N[(t$95$3 * t$95$0 + N[(t$95$1 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -1.05e-17], t$95$4, If[LessEqual[phi2, 10400000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.04999999999999996e-17 or 1.04e7 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   11. Applied rewrites 57.1%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \left(\phi_2 - \phi_1\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \color{blue}{\left(\phi_2 - \phi_1\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       6. *-commutative N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       7. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       8. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       9. lower-+.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       10. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       11. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       12. *-commutative N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       14. lower-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2 \cdot \sin \phi_1}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       15. lower-sin.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2} \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       16. lower-sin.f64 58.1%

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \color{blue}{\sin \phi_1}, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   13. Applied rewrites 58.1%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]
   14. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       6. *-commutative N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       7. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       8. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       9. lower-+.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       10. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       11. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       12. *-commutative N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       14. lower-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       15. lower-sin.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       16. lower-sin.f64 73.6%

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   15. Applied rewrites 73.6%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   if -1.04999999999999996e-17 < phi2 < 1.04e7

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 63.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   14. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

   15. Applied rewrites 61.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := t\_0 + \sin \phi_2 \cdot \sin \phi_1\\ t_2 := \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_3 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\\ t_4 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_3, t\_0, \mathsf{fma}\left(t\_1, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - t\_1 \cdot -0.5\right) - t\_3 \cdot t\_0}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -0.25:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\phi_1 \leq 2.4 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (+ t_0 (* (sin phi2) (sin phi1))))
        (t_2
         (fma
          (cos phi2)
          (pow
           (fma
            (cos (* -0.5 lambda2))
            (sin (* 0.5 lambda1))
            (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
           2.0)
          (pow (sin (* -0.5 phi2)) 2.0)))
        (t_3 (fma -0.5 (cos (- lambda2 lambda1)) 0.5))
        (t_4
         (*
          (*
           (atan2
            (sqrt (fma t_3 t_0 (fma t_1 -0.5 0.5)))
            (sqrt (- (- 0.5 (* t_1 -0.5)) (* t_3 t_0))))
           2.0)
          R)))
   (if (<= phi1 -0.25)
     t_4
     (if (<= phi1 2.4e-15)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = t_0 + (sin(phi2) * sin(phi1));
	double t_2 = fma(cos(phi2), pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0), pow(sin((-0.5 * phi2)), 2.0));
	double t_3 = fma(-0.5, cos((lambda2 - lambda1)), 0.5);
	double t_4 = (atan2(sqrt(fma(t_3, t_0, fma(t_1, -0.5, 0.5))), sqrt(((0.5 - (t_1 * -0.5)) - (t_3 * t_0)))) * 2.0) * R;
	double tmp;
	if (phi1 <= -0.25) {
		tmp = t_4;
	} else if (phi1 <= 2.4e-15) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = Float64(t_0 + Float64(sin(phi2) * sin(phi1)))
	t_2 = fma(cos(phi2), (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0), (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_3 = fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)
	t_4 = Float64(Float64(atan(sqrt(fma(t_3, t_0, fma(t_1, -0.5, 0.5))), sqrt(Float64(Float64(0.5 - Float64(t_1 * -0.5)) - Float64(t_3 * t_0)))) * 2.0) * R)
	tmp = 0.0
	if (phi1 <= -0.25)
		tmp = t_4;
	elseif (phi1 <= 2.4e-15)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[ArcTan[N[Sqrt[N[(t$95$3 * t$95$0 + N[(t$95$1 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -0.25], t$95$4, If[LessEqual[phi1, 2.4e-15], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.25 or 2.39999999999999995e-15 < phi1

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   11. Applied rewrites 57.1%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \left(\phi_2 - \phi_1\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \color{blue}{\left(\phi_2 - \phi_1\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       6. *-commutative N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       7. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       8. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       9. lower-+.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       10. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       11. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       12. *-commutative N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       14. lower-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2 \cdot \sin \phi_1}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       15. lower-sin.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2} \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       16. lower-sin.f64 58.1%

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \color{blue}{\sin \phi_1}, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   13. Applied rewrites 58.1%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]
   14. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       6. *-commutative N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       7. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       8. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       9. lower-+.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       10. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       11. lift-cos.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       12. *-commutative N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       14. lower-*.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       15. lower-sin.f64 N/A

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       16. lower-sin.f64 73.6%

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   15. Applied rewrites 73.6%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   if -0.25 < phi1 < 2.39999999999999995e-15

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 86.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_3 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\\ t_4 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_3, t\_0, \mathsf{fma}\left(t\_1, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - t\_1 \cdot -0.5\right) - t\_3 \cdot t\_0}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -0.25:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\phi_1 \leq 2.4 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (fma (cos phi2) (cos phi1) (* (sin phi2) (sin phi1))))
        (t_2
         (fma
          (cos phi2)
          (pow
           (fma
            (cos (* -0.5 lambda2))
            (sin (* 0.5 lambda1))
            (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
           2.0)
          (pow (sin (* -0.5 phi2)) 2.0)))
        (t_3 (fma -0.5 (cos (- lambda2 lambda1)) 0.5))
        (t_4
         (*
          (*
           (atan2
            (sqrt (fma t_3 t_0 (fma t_1 -0.5 0.5)))
            (sqrt (- (- 0.5 (* t_1 -0.5)) (* t_3 t_0))))
           2.0)
          R)))
   (if (<= phi1 -0.25)
     t_4
     (if (<= phi1 2.4e-15)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = fma(cos(phi2), cos(phi1), (sin(phi2) * sin(phi1)));
	double t_2 = fma(cos(phi2), pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0), pow(sin((-0.5 * phi2)), 2.0));
	double t_3 = fma(-0.5, cos((lambda2 - lambda1)), 0.5);
	double t_4 = (atan2(sqrt(fma(t_3, t_0, fma(t_1, -0.5, 0.5))), sqrt(((0.5 - (t_1 * -0.5)) - (t_3 * t_0)))) * 2.0) * R;
	double tmp;
	if (phi1 <= -0.25) {
		tmp = t_4;
	} else if (phi1 <= 2.4e-15) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = fma(cos(phi2), cos(phi1), Float64(sin(phi2) * sin(phi1)))
	t_2 = fma(cos(phi2), (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0), (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_3 = fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)
	t_4 = Float64(Float64(atan(sqrt(fma(t_3, t_0, fma(t_1, -0.5, 0.5))), sqrt(Float64(Float64(0.5 - Float64(t_1 * -0.5)) - Float64(t_3 * t_0)))) * 2.0) * R)
	tmp = 0.0
	if (phi1 <= -0.25)
		tmp = t_4;
	elseif (phi1 <= 2.4e-15)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[ArcTan[N[Sqrt[N[(t$95$3 * t$95$0 + N[(t$95$1 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -0.25], t$95$4, If[LessEqual[phi1, 2.4e-15], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.25 or 2.39999999999999995e-15 < phi1

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   11. Applied rewrites 57.1%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R} \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \left(\phi_2 - \phi_1\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \color{blue}{\left(\phi_2 - \phi_1\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}, \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       8. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \left(\phi_2 - \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 58.1%

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   13. Applied rewrites 58.1%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)}, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]
   14. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       2. lift--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       3. cos-diff N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       4. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       5. lift-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       6. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       7. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       8. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right), \frac{-1}{2}, \frac{1}{2}\right)\right)}}{\sqrt{\left(\frac{1}{2} - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right) \cdot \frac{-1}{2}\right) - \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 73.6%

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   15. Applied rewrites 73.6%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, \sin \phi_2 \cdot \sin \phi_1\right)} \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   if -0.25 < phi1 < 2.39999999999999995e-15

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(-0.5 \cdot \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot t\_0\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(t\_0, \cos \left(\lambda_1 \cdot 0.5\right), \sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -70000000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 2.1 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* -0.5 lambda2)))
        (t_1
         (fma
          (cos phi1)
          (pow
           (fma
            (cos (* -0.5 lambda2))
            (sin (* 0.5 lambda1))
            (* (cos (* 0.5 lambda1)) t_0))
           2.0)
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_2
         (fma
          (cos phi2)
          (pow
           (fma
            t_0
            (cos (* lambda1 0.5))
            (* (sin (* lambda1 0.5)) (cos (* lambda2 0.5))))
           2.0)
          (pow (sin (* 0.5 phi2)) 2.0)))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -70000000000000.0)
     t_3
     (if (<= phi2 2.1e-13)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((-0.5 * lambda2));
	double t_1 = fma(cos(phi1), pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * t_0)), 2.0), pow(sin((0.5 * phi1)), 2.0));
	double t_2 = fma(cos(phi2), pow(fma(t_0, cos((lambda1 * 0.5)), (sin((lambda1 * 0.5)) * cos((lambda2 * 0.5)))), 2.0), pow(sin((0.5 * phi2)), 2.0));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -70000000000000.0) {
		tmp = t_3;
	} else if (phi2 <= 2.1e-13) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(-0.5 * lambda2))
	t_1 = fma(cos(phi1), (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * t_0)) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_2 = fma(cos(phi2), (fma(t_0, cos(Float64(lambda1 * 0.5)), Float64(sin(Float64(lambda1 * 0.5)) * cos(Float64(lambda2 * 0.5)))) ^ 2.0), (sin(Float64(0.5 * phi2)) ^ 2.0))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -70000000000000.0)
		tmp = t_3;
	elseif (phi2 <= 2.1e-13)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(t$95$0 * N[Cos[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(lambda2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -70000000000000.0], t$95$3, If[LessEqual[phi2, 2.1e-13], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -7e13 or 2.09999999999999989e-13 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\lambda_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\lambda_2 \cdot \frac{1}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       10. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       12. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       13. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       15. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       16. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       18. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\mathsf{neg}\left(\frac{\lambda_2}{-2}\right)\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       19. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\mathsf{neg}\left(\lambda_2 \cdot \frac{1}{-2}\right)\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\mathsf{neg}\left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot 0.5\right), \sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\lambda_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\lambda_2 \cdot \frac{1}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       10. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       12. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       13. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right) + \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       15. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       16. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       18. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\mathsf{neg}\left(\frac{\lambda_2}{-2}\right)\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       19. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\mathsf{neg}\left(\lambda_2 \cdot \frac{1}{-2}\right)\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\mathsf{neg}\left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right), \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot 0.5\right), \sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \lambda_2\right), \cos \left(\lambda_1 \cdot 0.5\right), \sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   if -7e13 < phi2 < 2.09999999999999989e-13

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -70000000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 10^{-15}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* -0.5 lambda2))
           (sin (* 0.5 lambda1))
           (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
          2.0))
        (t_1 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
        (t_2 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -70000000000000.0)
     t_3
     (if (<= phi2 1e-15)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0);
	double t_1 = fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0));
	double t_2 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -70000000000000.0) {
		tmp = t_3;
	} else if (phi2 <= 1e-15) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0
	t_1 = fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_2 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -70000000000000.0)
		tmp = t_3;
	elseif (phi2 <= 1e-15)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -70000000000000.0], t$95$3, If[LessEqual[phi2, 1e-15], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -7e13 or 1.0000000000000001e-15 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -7e13 < phi2 < 1.0000000000000001e-15

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 75.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -70000000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 580000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* -0.5 lambda2))
           (sin (* 0.5 lambda1))
           (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
          2.0))
        (t_1 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
        (t_2 (fma (cos phi2) t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* phi2 0.5)))))))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -70000000000000.0)
     t_3
     (if (<= phi2 580000000.0)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0);
	double t_1 = fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0));
	double t_2 = fma(cos(phi2), t_0, (0.5 - (0.5 * cos((2.0 * (phi2 * 0.5))))));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -70000000000000.0) {
		tmp = t_3;
	} else if (phi2 <= 580000000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0
	t_1 = fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_2 = fma(cos(phi2), t_0, Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(phi2 * 0.5))))))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -70000000000000.0)
		tmp = t_3;
	elseif (phi2 <= 580000000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(phi2 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -70000000000000.0], t$95$3, If[LessEqual[phi2, 580000000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -7e13 or 5.8e8 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. sqr-sin-a N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 55.1%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       12. lower-*.f64 55.1%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 55.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       5. sqr-sin-a N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}\right) \]

       9. lower-*.f64 55.1%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_2\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 55.1%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}\right) \]

   23. Applied rewrites 55.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}\right) \]

   if -7e13 < phi2 < 5.8e8

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 69.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)\\ t_3 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_1 \leq -0.25:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(t\_0, t\_3, 0.5 - 0.5 \cdot \left(t\_3 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, t\_3, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right)\\ \mathbf{elif}\;\phi_1 \leq 3.9 \cdot 10^{-82}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1}}{\sqrt{\left|\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - t\_0 \cdot t\_3\right|}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (fma
          (cos phi2)
          (pow
           (fma
            (cos (* -0.5 lambda2))
            (sin (* 0.5 lambda1))
            (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
           2.0)
          (- 0.5 (* 0.5 (cos (* 2.0 (* phi2 0.5)))))))
        (t_3 (* (cos phi2) (cos phi1))))
   (if (<= phi1 -0.25)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (pow
          (sqrt
           (fma t_0 t_3 (- 0.5 (* 0.5 (+ t_3 (* (sin phi2) (sin phi1)))))))
          2.0))
        (sqrt
         (-
          1.0
          (fma
           (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
           t_3
           (- 0.5 (* (cos (- phi2 phi1)) 0.5))))))))
     (if (<= phi1 3.9e-82)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (+
            (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
            (* (* (* (cos phi1) (cos phi2)) t_1) t_1)))
          (sqrt
           (fabs
            (-
             (+ 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
             (* t_0 t_3)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = fma(cos(phi2), pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0), (0.5 - (0.5 * cos((2.0 * (phi2 * 0.5))))));
	double t_3 = cos(phi2) * cos(phi1);
	double tmp;
	if (phi1 <= -0.25) {
		tmp = R * (2.0 * atan2(sqrt(pow(sqrt(fma(t_0, t_3, (0.5 - (0.5 * (t_3 + (sin(phi2) * sin(phi1))))))), 2.0)), sqrt((1.0 - fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), t_3, (0.5 - (cos((phi2 - phi1)) * 0.5)))))));
	} else if (phi1 <= 3.9e-82) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt(fabs(((0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) - (t_0 * t_3))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = fma(cos(phi2), (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(phi2 * 0.5))))))
	t_3 = Float64(cos(phi2) * cos(phi1))
	tmp = 0.0
	if (phi1 <= -0.25)
		tmp = Float64(R * Float64(2.0 * atan(sqrt((sqrt(fma(t_0, t_3, Float64(0.5 - Float64(0.5 * Float64(t_3 + Float64(sin(phi2) * sin(phi1))))))) ^ 2.0)), sqrt(Float64(1.0 - fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), t_3, Float64(0.5 - Float64(cos(Float64(phi2 - phi1)) * 0.5))))))));
	elseif (phi1 <= 3.9e-82)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt(abs(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))) - Float64(t_0 * t_3)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(phi2 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -0.25], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[Power[N[Sqrt[N[(t$95$0 * t$95$3 + N[(0.5 - N[(0.5 * N[(t$95$3 + N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * t$95$3 + N[(0.5 - N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 3.9e-82], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Abs[N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.25

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 57.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 57.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}{\sqrt{1 - \color{blue}{{\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      3. associate-*r* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(\left(2 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 - \phi_2\right)\right)}\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      4. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\color{blue}{1} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      5. *-lft-identity 57.0%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \color{blue}{\left(\phi_1 - \phi_2\right)}\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\phi_1 - \phi_2\right)}\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      7. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\phi_1 - \phi_2\right)\right)\right)}\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      8. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      9. sub-negate-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(\phi_2 - \phi_1\right)}\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      10. cos-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)}\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      12. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      13. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1\right)}\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      17. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      20. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

      21. lower-sin.f64 57.5%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]

   7. Applied rewrites 57.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)}\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\right)}^{2}}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right)}\right)}^{2}}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}\right)\right)}\right)}^{2}}}\right) \]

      3. associate-*r* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(\left(2 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 - \phi_2\right)\right)}\right)}\right)}^{2}}}\right) \]

      4. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\color{blue}{1} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right)}^{2}}}\right) \]

      5. *-lft-identity 57.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \color{blue}{\left(\phi_1 - \phi_2\right)}\right)}\right)}^{2}}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\phi_1 - \phi_2\right)}\right)}\right)}^{2}}}\right) \]

      7. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\phi_1 - \phi_2\right)\right)\right)}\right)}\right)}^{2}}}\right) \]

      8. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right)\right)}\right)}^{2}}}\right) \]

      9. sub-negate-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(\phi_2 - \phi_1\right)}\right)}\right)}^{2}}}\right) \]

      10. cos-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)}\right)}\right)}^{2}}}\right) \]

      11. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}\right) \]

      12. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}\right) \]

      13. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_2 \cdot \sin \phi_1\right)}\right)}\right)}^{2}}}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}\right) \]

      17. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos \phi_2 \cdot \cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}\right) \]

      19. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2 \cdot \sin \phi_1}\right)\right)}\right)}^{2}}}\right) \]

      20. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \color{blue}{\sin \phi_2} \cdot \sin \phi_1\right)\right)}\right)}^{2}}}\right) \]

      21. lower-sin.f64 73.4%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right)}\right)}^{2}}}\right) \]

   9. Applied rewrites 73.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)}\right)}\right)}^{2}}}\right) \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - \color{blue}{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}}\right) \]

       2. lift-sqrt.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\color{blue}{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}}^{2}}}\right) \]

       3. sqrt-pow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - \color{blue}{{\left(\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}^{\left(\frac{2}{2}\right)}}}}\right) \]

       4. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - {\left(\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)\right)}^{\color{blue}{1}}}}\right) \]

       5. unpow1 73.5%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}}}\right) \]

   11. Applied rewrites 58.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)\right)}\right)}^{2}}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]

   if -0.25 < phi1 < 3.89999999999999973e-82

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. sqr-sin-a N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 55.1%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       12. lower-*.f64 55.1%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 55.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}\right) \]

       5. sqr-sin-a N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}\right) \]

       9. lower-*.f64 55.1%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_2\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 55.1%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}\right) \]

   23. Applied rewrites 55.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}\right) \]

   if 3.89999999999999973e-82 < phi1

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\left|\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right|}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 63.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot t\_0\\ t_2 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_1, t\_0, t\_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.235:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{elif}\;\phi_1 \leq 2.4 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, t\_2\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi2) t_0)))
        (t_2 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1))))))
        (t_3 (fma (cos phi1) t_0 t_2)))
   (if (<= phi1 -0.235)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (if (<= phi1 2.4e-15)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
          (sqrt
           (-
            1.0
            (fma
             (fma -0.5 (cos (- lambda2 lambda1)) 0.5)
             (cos phi1)
             t_2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi2) * t_0);
	double t_2 = 0.5 - (0.5 * cos((2.0 * (0.5 * phi1))));
	double t_3 = fma(cos(phi1), t_0, t_2);
	double tmp;
	if (phi1 <= -0.235) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else if (phi1 <= 2.4e-15) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0))), sqrt((1.0 - fma(fma(-0.5, cos((lambda2 - lambda1)), 0.5), cos(phi1), t_2)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi2) * t_0))
	t_2 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))
	t_3 = fma(cos(phi1), t_0, t_2)
	tmp = 0.0
	if (phi1 <= -0.235)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	elseif (phi1 <= 2.4e-15)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(1.0 - fma(fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5), cos(phi1), t_2))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision]}, If[LessEqual[phi1, -0.235], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 2.4e-15], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.23499999999999999

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. sqr-sin-a N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      11. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      12. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\phi_1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\phi_1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      14. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      15. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      16. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      17. lift-*.f64 45.2%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 45.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}\right) \]

       5. sqr-sin-a N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       9. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       11. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       12. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\phi_1}{2}\right)\right)}}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\phi_1}{2}\right)\right)}}\right) \]

       14. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       16. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       17. lift-*.f64 45.2%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Applied rewrites 45.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   if -0.23499999999999999 < phi1 < 2.39999999999999995e-15

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 53.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 53.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 51.0%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 51.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 2.39999999999999995e-15 < phi1

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 62.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right), {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\phi_2 \leq -2100000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 10^{-15}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))))
        (t_1
         (fma
          (cos phi2)
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda2 lambda1) -0.5)))))
          (pow (sin (* 0.5 phi2)) 2.0)))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= phi2 -2100000000000.0)
     t_2
     (if (<= phi2 1e-15)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double t_1 = fma(cos(phi2), (0.5 - (0.5 * cos((2.0 * ((lambda2 - lambda1) * -0.5))))), pow(sin((0.5 * phi2)), 2.0));
	double t_2 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double tmp;
	if (phi2 <= -2100000000000.0) {
		tmp = t_2;
	} else if (phi2 <= 1e-15) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	t_1 = fma(cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda2 - lambda1) * -0.5))))), (sin(Float64(0.5 * phi2)) ^ 2.0))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	tmp = 0.0
	if (phi2 <= -2100000000000.0)
		tmp = t_2;
	elseif (phi2 <= 1e-15)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2100000000000.0], t$95$2, If[LessEqual[phi2, 1e-15], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.1e12 or 1.0000000000000001e-15 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 44.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 44.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right), {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, 0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   if -2.1e12 < phi2 < 1.0000000000000001e-15

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 53.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 53.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 51.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 51.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 62.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left|\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)\right|\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := t\_2 + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_4 := t\_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}} \leq 0.4:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fabs
          (fma
           (fma -0.5 (cos (- lambda2 lambda1)) 0.5)
           (* (cos phi2) (cos phi1))
           (fma (cos (- phi2 phi1)) -0.5 0.5))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_3
         (+ t_2 (* (cos phi2) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))))
        (t_4 (+ t_2 (* (* (* (cos phi1) (cos phi2)) t_1) t_1))))
   (if (<= (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))) 0.4)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fabs(fma(fma(-0.5, cos((lambda2 - lambda1)), 0.5), (cos(phi2) * cos(phi1)), fma(cos((phi2 - phi1)), -0.5, 0.5)));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_3 = t_2 + (cos(phi2) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double t_4 = t_2 + (((cos(phi1) * cos(phi2)) * t_1) * t_1);
	double tmp;
	if ((2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4)))) <= 0.4) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = abs(fma(fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5), Float64(cos(phi2) * cos(phi1)), fma(cos(Float64(phi2 - phi1)), -0.5, 0.5)))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_3 = Float64(t_2 + Float64(cos(phi2) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	t_4 = Float64(t_2 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))) <= 0.4)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Abs[N[(N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.4], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))))))) < 0.40000000000000002

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 53.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 53.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 51.0%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 51.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 0.40000000000000002 < (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))))))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   11. Applied rewrites 58.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left|\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)\right|}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left|\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)\right|}}{\sqrt{1 - \color{blue}{\left|\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)\right|}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 62.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \cos \left(\phi_2 - \phi_1\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\\ t_5 := t\_3 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;t\_3 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2 \leq 0.015:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_4, t\_0, \mathsf{fma}\left(t\_1, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - t\_1 \cdot -0.5\right) - t\_4 \cdot t\_0}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (cos (- phi2 phi1)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_4 (fma -0.5 (cos (- lambda2 lambda1)) 0.5))
        (t_5
         (+ t_3 (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))))
   (if (<= (+ t_3 (* (* (* (cos phi1) (cos phi2)) t_2) t_2)) 0.015)
     (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
     (*
      (*
       (atan2
        (sqrt (fma t_4 t_0 (fma t_1 -0.5 0.5)))
        (sqrt (- (- 0.5 (* t_1 -0.5)) (* t_4 t_0))))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = cos((phi2 - phi1));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_4 = fma(-0.5, cos((lambda2 - lambda1)), 0.5);
	double t_5 = t_3 + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double tmp;
	if ((t_3 + (((cos(phi1) * cos(phi2)) * t_2) * t_2)) <= 0.015) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = (atan2(sqrt(fma(t_4, t_0, fma(t_1, -0.5, 0.5))), sqrt(((0.5 - (t_1 * -0.5)) - (t_4 * t_0)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = cos(Float64(phi2 - phi1))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_4 = fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)
	t_5 = Float64(t_3 + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	tmp = 0.0
	if (Float64(t_3 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2)) <= 0.015)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(t_4, t_0, fma(t_1, -0.5, 0.5))), sqrt(Float64(Float64(0.5 - Float64(t_1 * -0.5)) - Float64(t_4 * t_0)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$3 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 0.015], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$4 * t$95$0 + N[(t$95$1 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] - N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 0.014999999999999999

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 53.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 53.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 51.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 51.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 0.014999999999999999 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   11. Applied rewrites 57.1%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 62.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0}}{\sqrt{\left|\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right|}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
        (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
      (sqrt
       (fabs
        (-
         (+ 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
         (*
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
          (* (cos phi2) (cos phi1)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(fabs(((0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1))))))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    code = r * (2.0d0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(abs(((0.5d0 + (0.5d0 * cos((2.0d0 * (0.5d0 * (phi1 - phi2)))))) - ((0.5d0 - (0.5d0 * cos((2.0d0 * (0.5d0 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1))))))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * t_0) * t_0))), Math.sqrt(Math.abs(((0.5 + (0.5 * Math.cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * Math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (Math.cos(phi2) * Math.cos(phi1))))))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	return R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0))), math.sqrt(math.fabs(((0.5 + (0.5 * math.cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (math.cos(phi2) * math.cos(phi1))))))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(abs(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))) - Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))) * Float64(cos(phi2) * cos(phi1)))))))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	tmp = R * (2.0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(abs(((0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1))))))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Abs[N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 62.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\left|\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right|}}}\right) \]
4. Add Preprocessing

Alternative 20: 62.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
        (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
      (sqrt
       (-
        (+ 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
        (*
         (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
         (* (cos phi2) (cos phi1))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(((0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1)))))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    code = r * (2.0d0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(((0.5d0 + (0.5d0 * cos((2.0d0 * (0.5d0 * (phi1 - phi2)))))) - ((0.5d0 - (0.5d0 * cos((2.0d0 * (0.5d0 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1)))))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * t_0) * t_0))), Math.sqrt(((0.5 + (0.5 * Math.cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * Math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (Math.cos(phi2) * Math.cos(phi1)))))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	return R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0))), math.sqrt(((0.5 + (0.5 * math.cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (math.cos(phi2) * math.cos(phi1)))))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))) - Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))) * Float64(cos(phi2) * cos(phi1))))))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	tmp = R * (2.0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(((0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1)))))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 62.3%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\color{blue}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}}\right) \]
4. Add Preprocessing

Alternative 21: 61.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_3 := \sqrt{t\_2}\\ \mathbf{if}\;\phi_1 \leq -4 \cdot 10^{-35}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_3}{\sqrt{1 - t\_2}}\right)\\ \mathbf{elif}\;\phi_1 \leq 2.4 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_3}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))
        (t_2 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
        (t_3 (sqrt t_2)))
   (if (<= phi1 -4e-35)
     (* R (* 2.0 (atan2 t_3 (sqrt (- 1.0 t_2)))))
     (if (<= phi1 2.4e-15)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       (*
        R
        (*
         2.0
         (atan2
          t_3
          (sqrt
           (-
            1.0
            (fma
             (fma -0.5 (cos (- lambda2 lambda1)) 0.5)
             (cos phi1)
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double t_2 = fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0));
	double t_3 = sqrt(t_2);
	double tmp;
	if (phi1 <= -4e-35) {
		tmp = R * (2.0 * atan2(t_3, sqrt((1.0 - t_2))));
	} else if (phi1 <= 2.4e-15) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * atan2(t_3, sqrt((1.0 - fma(fma(-0.5, cos((lambda2 - lambda1)), 0.5), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_2 = fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_3 = sqrt(t_2)
	tmp = 0.0
	if (phi1 <= -4e-35)
		tmp = Float64(R * Float64(2.0 * atan(t_3, sqrt(Float64(1.0 - t_2)))));
	elseif (phi1 <= 2.4e-15)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_3, sqrt(Float64(1.0 - fma(fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, If[LessEqual[phi1, -4e-35], N[(R * N[(2.0 * N[ArcTan[t$95$3 / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 2.4e-15], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[t$95$3 / N[Sqrt[N[(1.0 - N[(N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -4.00000000000000003e-35

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if -4.00000000000000003e-35 < phi1 < 2.39999999999999995e-15

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.6%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if 2.39999999999999995e-15 < phi1

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 61.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right), {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)\\ t_1 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{if}\;\phi_2 \leq -1.05 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 10^{-15}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi2)
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda2 lambda1) -0.5)))))
          (pow (sin (* 0.5 phi2)) 2.0)))
        (t_1 (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))))
   (if (<= phi2 -1.05e-18)
     t_1
     (if (<= phi2 1e-15)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
            (pow (sin (* 0.5 phi1)) 2.0)))
          (sqrt
           (-
            1.0
            (fma
             (fma -0.5 (cos (- lambda2 lambda1)) 0.5)
             (cos phi1)
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))))))
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi2), (0.5 - (0.5 * cos((2.0 * ((lambda2 - lambda1) * -0.5))))), pow(sin((0.5 * phi2)), 2.0));
	double t_1 = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	double tmp;
	if (phi2 <= -1.05e-18) {
		tmp = t_1;
	} else if (phi2 <= 1e-15) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((0.5 * phi1)), 2.0))), sqrt((1.0 - fma(fma(-0.5, cos((lambda2 - lambda1)), 0.5), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda2 - lambda1) * -0.5))))), (sin(Float64(0.5 * phi2)) ^ 2.0))
	t_1 = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
	tmp = 0.0
	if (phi2 <= -1.05e-18)
		tmp = t_1;
	elseif (phi2 <= 1e-15)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(1.0 - fma(fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.05e-18], t$95$1, If[LessEqual[phi2, 1e-15], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.05e-18 or 1.0000000000000001e-15 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 44.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 44.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right), {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, 0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   if -1.05e-18 < phi2 < 1.0000000000000001e-15

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 61.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)\\ t_1 := \left(\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -70000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 580000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda2 lambda1) -0.5)))))
          (cos phi2)
          (- 0.5 (* 0.5 (cos (* 2.0 (* phi2 0.5)))))))
        (t_1 (* (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))) 2.0) R)))
   (if (<= phi2 -70000000000000.0)
     t_1
     (if (<= phi2 580000000.0)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
            (pow (sin (* 0.5 phi1)) 2.0)))
          (sqrt
           (-
            1.0
            (fma
             (fma -0.5 (cos (- lambda2 lambda1)) 0.5)
             (cos phi1)
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))))))
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((0.5 - (0.5 * cos((2.0 * ((lambda2 - lambda1) * -0.5))))), cos(phi2), (0.5 - (0.5 * cos((2.0 * (phi2 * 0.5))))));
	double t_1 = (atan2(sqrt(t_0), sqrt((1.0 - t_0))) * 2.0) * R;
	double tmp;
	if (phi2 <= -70000000000000.0) {
		tmp = t_1;
	} else if (phi2 <= 580000000.0) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((0.5 * phi1)), 2.0))), sqrt((1.0 - fma(fma(-0.5, cos((lambda2 - lambda1)), 0.5), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda2 - lambda1) * -0.5))))), cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(phi2 * 0.5))))))
	t_1 = Float64(Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))) * 2.0) * R)
	tmp = 0.0
	if (phi2 <= -70000000000000.0)
		tmp = t_1;
	elseif (phi2 <= 580000000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(1.0 - fma(fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(phi2 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -70000000000000.0], t$95$1, If[LessEqual[phi2, 580000000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -7e13 or 5.8e8 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 42.5%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}} \cdot 2\right) \cdot R} \]

   if -7e13 < phi2 < 5.8e8

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 58.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_1 := \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -70000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 580000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (fma (cos (- lambda2 lambda1)) -0.5 0.5)
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_1
         (fma
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda2 lambda1) -0.5)))))
          (cos phi2)
          (- 0.5 (* 0.5 (cos (* 2.0 (* phi2 0.5)))))))
        (t_2 (* (* (atan2 (sqrt t_1) (sqrt (- 1.0 t_1))) 2.0) R)))
   (if (<= phi2 -70000000000000.0)
     t_2
     (if (<= phi2 580000000.0)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi1), fma(cos((lambda2 - lambda1)), -0.5, 0.5), pow(sin((0.5 * phi1)), 2.0));
	double t_1 = fma((0.5 - (0.5 * cos((2.0 * ((lambda2 - lambda1) * -0.5))))), cos(phi2), (0.5 - (0.5 * cos((2.0 * (phi2 * 0.5))))));
	double t_2 = (atan2(sqrt(t_1), sqrt((1.0 - t_1))) * 2.0) * R;
	double tmp;
	if (phi2 <= -70000000000000.0) {
		tmp = t_2;
	} else if (phi2 <= 580000000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi1), fma(cos(Float64(lambda2 - lambda1)), -0.5, 0.5), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_1 = fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda2 - lambda1) * -0.5))))), cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(phi2 * 0.5))))))
	t_2 = Float64(Float64(atan(sqrt(t_1), sqrt(Float64(1.0 - t_1))) * 2.0) * R)
	tmp = 0.0
	if (phi2 <= -70000000000000.0)
		tmp = t_2;
	elseif (phi2 <= 580000000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(phi2 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -70000000000000.0], t$95$2, If[LessEqual[phi2, 580000000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -7e13 or 5.8e8 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 78.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. mult-flip-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       22. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       23. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       25. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   21. Applied rewrites 42.5%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_2 \cdot 0.5\right)\right)\right)}} \cdot 2\right) \cdot R} \]

   if -7e13 < phi2 < 5.8e8

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. sqr-sin-a-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \color{blue}{\frac{1}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{-1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \frac{-1}{2} + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      12. lower-fma.f64 43.8%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 43.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       5. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       8. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       9. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \color{blue}{\frac{1}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{-1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \frac{-1}{2} + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       12. lower-fma.f64 43.8%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 43.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 57.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\phi_2 - \phi_1\right)\\ t_1 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\\ t_2 := \mathsf{fma}\left(t\_1, \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ t_3 := \left(\tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{-10}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_1 \leq 2.4 \cdot 10^{-15}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, \cos \phi_2, \mathsf{fma}\left(t\_0, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - t\_0 \cdot -0.5\right) - t\_1 \cdot \cos \phi_2}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- phi2 phi1)))
        (t_1 (fma -0.5 (cos (- lambda2 lambda1)) 0.5))
        (t_2 (fma t_1 (cos phi1) (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
        (t_3 (* (* (atan2 (sqrt t_2) (sqrt (- 1.0 t_2))) 2.0) R)))
   (if (<= phi1 -2.8e-10)
     t_3
     (if (<= phi1 2.4e-15)
       (*
        (*
         (atan2
          (sqrt (fma t_1 (cos phi2) (fma t_0 -0.5 0.5)))
          (sqrt (- (- 0.5 (* t_0 -0.5)) (* t_1 (cos phi2)))))
         2.0)
        R)
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((phi2 - phi1));
	double t_1 = fma(-0.5, cos((lambda2 - lambda1)), 0.5);
	double t_2 = fma(t_1, cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))));
	double t_3 = (atan2(sqrt(t_2), sqrt((1.0 - t_2))) * 2.0) * R;
	double tmp;
	if (phi1 <= -2.8e-10) {
		tmp = t_3;
	} else if (phi1 <= 2.4e-15) {
		tmp = (atan2(sqrt(fma(t_1, cos(phi2), fma(t_0, -0.5, 0.5))), sqrt(((0.5 - (t_0 * -0.5)) - (t_1 * cos(phi2))))) * 2.0) * R;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(phi2 - phi1))
	t_1 = fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)
	t_2 = fma(t_1, cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	t_3 = Float64(Float64(atan(sqrt(t_2), sqrt(Float64(1.0 - t_2))) * 2.0) * R)
	tmp = 0.0
	if (phi1 <= -2.8e-10)
		tmp = t_3;
	elseif (phi1 <= 2.4e-15)
		tmp = Float64(Float64(atan(sqrt(fma(t_1, cos(phi2), fma(t_0, -0.5, 0.5))), sqrt(Float64(Float64(0.5 - Float64(t_0 * -0.5)) - Float64(t_1 * cos(phi2))))) * 2.0) * R);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -2.8e-10], t$95$3, If[LessEqual[phi1, 2.4e-15], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$1 * N[Cos[phi2], $MachinePrecision] + N[(t$95$0 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.80000000000000015e-10 or 2.39999999999999995e-15 < phi1

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 42.6%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}} \cdot 2\right) \cdot R} \]

   if -2.80000000000000015e-10 < phi1 < 2.39999999999999995e-15

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   11. Applied rewrites 57.1%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R} \]

   12. Taylor expanded in phi1 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \color{blue}{\cos \phi_2}, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]
   13. Step-by-step derivation

       1. lower-cos.f64 48.0%

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   14. Applied rewrites 48.0%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \color{blue}{\cos \phi_2}, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   15. Taylor expanded in phi1 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \color{blue}{\cos \phi_2}}} \cdot 2\right) \cdot R \]
   16. Step-by-step derivation

       1. lower-cos.f64 45.8%

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \cos \phi_2}} \cdot 2\right) \cdot R \]

   17. Applied rewrites 45.8%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \color{blue}{\cos \phi_2}}} \cdot 2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 49.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\\ t_2 := \cos \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)}^{0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, \cos \phi_2, \mathsf{fma}\left(t\_2, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - t\_2 \cdot -0.5\right) - t\_1 \cdot \cos \phi_2}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (fma -0.5 (cos (- lambda2 lambda1)) 0.5))
        (t_2 (cos (- phi2 phi1))))
   (if (<=
        (+
         (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
         (* (* (* (cos phi1) (cos phi2)) t_0) t_0))
        5e-15)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (fma
          (cos phi1)
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (pow (* 0.5 phi1) 2.0)))
        (pow
         (-
          1.0
          (fma
           (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
           (cos phi1)
           (* (* phi1 0.5) (* phi1 0.5))))
         0.5))))
     (*
      (*
       (atan2
        (sqrt (fma t_1 (cos phi2) (fma t_2 -0.5 0.5)))
        (sqrt (- (- 0.5 (* t_2 -0.5)) (* t_1 (cos phi2)))))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = fma(-0.5, cos((lambda2 - lambda1)), 0.5);
	double t_2 = cos((phi2 - phi1));
	double tmp;
	if ((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 5e-15) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow((0.5 * phi1), 2.0))), pow((1.0 - fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), ((phi1 * 0.5) * (phi1 * 0.5)))), 0.5)));
	} else {
		tmp = (atan2(sqrt(fma(t_1, cos(phi2), fma(t_2, -0.5, 0.5))), sqrt(((0.5 - (t_2 * -0.5)) - (t_1 * cos(phi2))))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)
	t_2 = cos(Float64(phi2 - phi1))
	tmp = 0.0
	if (Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 5e-15)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (Float64(0.5 * phi1) ^ 2.0))), (Float64(1.0 - fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), Float64(Float64(phi1 * 0.5) * Float64(phi1 * 0.5)))) ^ 0.5))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(t_1, cos(phi2), fma(t_2, -0.5, 0.5))), sqrt(Float64(Float64(0.5 - Float64(t_2 * -0.5)) - Float64(t_1 * cos(phi2))))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 5e-15], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(0.5 * phi1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[(1.0 - N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$1 * N[Cos[phi2], $MachinePrecision] + N[(t$95$2 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(t$95$2 * -0.5), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 4.99999999999999999e-15

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 31.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 21.9%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   14. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]

       2. pow1/2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)\right)}^{\frac{1}{2}}}}\right) \]

       3. lower-pow.f64 27.1%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)\right)}^{0.5}}}\right) \]

   15. Applied rewrites 27.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)}^{0.5}}}\right) \]

   if 4.99999999999999999e-15 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 77.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   11. Applied rewrites 57.1%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R} \]

   12. Taylor expanded in phi1 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \color{blue}{\cos \phi_2}, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]
   13. Step-by-step derivation

       1. lower-cos.f64 48.0%

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   14. Applied rewrites 48.0%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \color{blue}{\cos \phi_2}, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R \]

   15. Taylor expanded in phi1 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \color{blue}{\cos \phi_2}}} \cdot 2\right) \cdot R \]
   16. Step-by-step derivation

       1. lower-cos.f64 45.8%

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \cos \phi_2}} \cdot 2\right) \cdot R \]

   17. Applied rewrites 45.8%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \color{blue}{\cos \phi_2}}} \cdot 2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 40.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R\\ \mathbf{if}\;t\_1 \leq -0.06:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-16}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (*
          (*
           (atan2
            (sqrt
             (fma
              (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
              (cos phi1)
              (* (* phi1 0.5) (* phi1 0.5))))
            (sqrt
             (-
              1.0
              (fma
               (cos phi2)
               (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
               (pow (sin (* -0.5 phi2)) 2.0)))))
           2.0)
          R)))
   (if (<= t_1 -0.06)
     t_2
     (if (<= t_1 5e-16)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (phi1 - phi2))), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = (atan2(sqrt(fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), ((phi1 * 0.5) * (phi1 * 0.5)))), sqrt((1.0 - fma(cos(phi2), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((-0.5 * phi2)), 2.0))))) * 2.0) * R;
	double tmp;
	if (t_1 <= -0.06) {
		tmp = t_2;
	} else if (t_1 <= 5e-16) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), Float64(Float64(phi1 * 0.5) * Float64(phi1 * 0.5)))), sqrt(Float64(1.0 - fma(cos(phi2), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(-0.5 * phi2)) ^ 2.0))))) * 2.0) * R)
	tmp = 0.0
	if (t_1 <= -0.06)
		tmp = t_2;
	elseif (t_1 <= 5e-16)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[t$95$1, -0.06], t$95$2, If[LessEqual[t$95$1, 5e-16], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.059999999999999998 or 5.0000000000000004e-16 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 31.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 21.9%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.3%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}} \cdot 2\right) \cdot R} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]
   17. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       3. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       4. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       5. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       7. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       8. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       10. lower-*.f64 29.1%

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

   18. Applied rewrites 29.1%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]

   if -0.059999999999999998 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 5.0000000000000004e-16

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \phi_2\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-PI.f64 52.6%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\pi}, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 52.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \phi_2\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-PI.f64 51.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\pi}, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 51.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left({\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \color{blue}{\sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \color{blue}{\left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\color{blue}{\phi_2} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 39.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left({\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \color{blue}{\sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \color{blue}{\left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\color{blue}{\phi_2} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-PI.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower-pow.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 40.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.0%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 29.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.7%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 29.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 40.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R\\ t_2 := {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\\ \mathbf{if}\;t\_0 \leq -0.06:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-16}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (*
          (*
           (atan2
            (sqrt
             (fma
              (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
              (cos phi1)
              (* (* phi1 0.5) (* phi1 0.5))))
            (sqrt
             (-
              1.0
              (fma
               (cos phi1)
               (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
               (pow (sin (* 0.5 phi1)) 2.0)))))
           2.0)
          R))
        (t_2 (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))
   (if (<= t_0 -0.06)
     t_1
     (if (<= t_0 5e-16)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = (atan2(sqrt(fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), ((phi1 * 0.5) * (phi1 * 0.5)))), sqrt((1.0 - fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((0.5 * phi1)), 2.0))))) * 2.0) * R;
	double t_2 = pow(sin((0.5 * (phi1 - phi2))), 2.0);
	double tmp;
	if (t_0 <= -0.06) {
		tmp = t_1;
	} else if (t_0 <= 5e-16) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), Float64(Float64(phi1 * 0.5) * Float64(phi1 * 0.5)))), sqrt(Float64(1.0 - fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))))) * 2.0) * R)
	t_2 = sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0
	tmp = 0.0
	if (t_0 <= -0.06)
		tmp = t_1;
	elseif (t_0 <= 5e-16)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$0, -0.06], t$95$1, If[LessEqual[t$95$0, 5e-16], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.059999999999999998 or 5.0000000000000004e-16 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 31.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 21.9%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.3%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}} \cdot 2\right) \cdot R} \]

   16. Taylor expanded in phi2 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]
   17. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       3. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       4. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       5. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       7. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       8. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       10. lower-*.f64 28.9%

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

   18. Applied rewrites 28.9%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]

   if -0.059999999999999998 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 5.0000000000000004e-16

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \phi_2\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-PI.f64 52.6%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\pi}, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 52.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \phi_2\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-PI.f64 51.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\pi}, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 51.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left({\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \color{blue}{\sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \color{blue}{\left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\color{blue}{\phi_2} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 39.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left({\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \color{blue}{\sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \color{blue}{\left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\color{blue}{\phi_2} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-PI.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower-pow.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 40.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.0%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 29.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.7%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 29.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 37.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\\ t_1 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{if}\;\phi_2 \leq -3.2 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 2.5 \cdot 10^{-35}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)}^{0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
        (t_1 (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))))
   (if (<= phi2 -3.2e-64)
     t_1
     (if (<= phi2 2.5e-35)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
            (pow (* 0.5 phi1) 2.0)))
          (pow
           (-
            1.0
            (fma
             (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
             (cos phi1)
             (* (* phi1 0.5) (* phi1 0.5))))
           0.5))))
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (phi1 - phi2))), 2.0);
	double t_1 = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	double tmp;
	if (phi2 <= -3.2e-64) {
		tmp = t_1;
	} else if (phi2 <= 2.5e-35) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow((0.5 * phi1), 2.0))), pow((1.0 - fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), ((phi1 * 0.5) * (phi1 * 0.5)))), 0.5)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0
	t_1 = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
	tmp = 0.0
	if (phi2 <= -3.2e-64)
		tmp = t_1;
	elseif (phi2 <= 2.5e-35)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (Float64(0.5 * phi1) ^ 2.0))), (Float64(1.0 - fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), Float64(Float64(phi1 * 0.5) * Float64(phi1 * 0.5)))) ^ 0.5))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -3.2e-64], t$95$1, If[LessEqual[phi2, 2.5e-35], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(0.5 * phi1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[(1.0 - N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.19999999999999975e-64 or 2.49999999999999982e-35 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \phi_2\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-PI.f64 52.6%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\pi}, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 52.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \phi_2\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-PI.f64 51.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\pi}, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 51.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left({\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \color{blue}{\sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \color{blue}{\left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\color{blue}{\phi_2} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 39.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left({\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \color{blue}{\sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \color{blue}{\left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\color{blue}{\phi_2} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-PI.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower-pow.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 40.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.0%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 29.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.7%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 29.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]

   if -3.19999999999999975e-64 < phi2 < 2.49999999999999982e-35

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 31.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 21.9%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   14. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]

       2. pow1/2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)\right)}^{\frac{1}{2}}}}\right) \]

       3. lower-pow.f64 27.1%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)\right)}^{0.5}}}\right) \]

   15. Applied rewrites 27.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)}^{0.5}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 30: 35.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right) \cdot \cos \phi_1\right)\right)}^{0.5}} \cdot 2\right) \cdot R\\ \mathbf{if}\;t\_0 \leq -0.06:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
        (t_2
         (*
          (*
           (atan2
            (sqrt
             (fma
              (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
              (cos phi1)
              (* (* phi1 0.5) (* phi1 0.5))))
            (pow
             (-
              1.0
              (fma
               (* phi1 0.5)
               (* phi1 0.5)
               (* (fma (cos (- lambda2 lambda1)) -0.5 0.5) (cos phi1))))
             0.5))
           2.0)
          R)))
   (if (<= t_0 -0.06)
     t_2
     (if (<= t_0 2e-8)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(sin((0.5 * (phi1 - phi2))), 2.0);
	double t_2 = (atan2(sqrt(fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), ((phi1 * 0.5) * (phi1 * 0.5)))), pow((1.0 - fma((phi1 * 0.5), (phi1 * 0.5), (fma(cos((lambda2 - lambda1)), -0.5, 0.5) * cos(phi1)))), 0.5)) * 2.0) * R;
	double tmp;
	if (t_0 <= -0.06) {
		tmp = t_2;
	} else if (t_0 <= 2e-8) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0
	t_2 = Float64(Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), Float64(Float64(phi1 * 0.5) * Float64(phi1 * 0.5)))), (Float64(1.0 - fma(Float64(phi1 * 0.5), Float64(phi1 * 0.5), Float64(fma(cos(Float64(lambda2 - lambda1)), -0.5, 0.5) * cos(phi1)))) ^ 0.5)) * 2.0) * R)
	tmp = 0.0
	if (t_0 <= -0.06)
		tmp = t_2;
	elseif (t_0 <= 2e-8)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[(1.0 - N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision] + N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[t$95$0, -0.06], t$95$2, If[LessEqual[t$95$0, 2e-8], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.059999999999999998 or 2e-8 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 31.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 21.9%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.3%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}} \cdot 2\right) \cdot R} \]

   17. Applied rewrites 24.6%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right) \cdot \cos \phi_1\right)\right)}^{0.5}}} \cdot 2\right) \cdot R \]

   if -0.059999999999999998 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 2e-8

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \phi_2\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-PI.f64 52.6%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\pi}, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 52.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \phi_2\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-PI.f64 51.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\pi}, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 51.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left({\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \color{blue}{\sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \color{blue}{\left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\color{blue}{\phi_2} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 39.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left({\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \color{blue}{\sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \color{blue}{\left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\color{blue}{\phi_2} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-PI.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower-pow.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 40.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.0%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 29.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.7%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 29.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 31: 31.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\\ \mathbf{if}\;\frac{\lambda_1 - \lambda_2}{2} \leq -500000:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
          (+ 1.0 (* -0.5 (pow phi1 2.0)))
          (* (* phi1 0.5) (* phi1 0.5))))
        (t_1 (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))
   (if (<= (/ (- lambda1 lambda2) 2.0) -500000.0)
     (* (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))) 2.0) R)
     (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), (1.0 + (-0.5 * pow(phi1, 2.0))), ((phi1 * 0.5) * (phi1 * 0.5)));
	double t_1 = pow(sin((0.5 * (phi1 - phi2))), 2.0);
	double tmp;
	if (((lambda1 - lambda2) / 2.0) <= -500000.0) {
		tmp = (atan2(sqrt(t_0), sqrt((1.0 - t_0))) * 2.0) * R;
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))), Float64(Float64(phi1 * 0.5) * Float64(phi1 * 0.5)))
	t_1 = sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(lambda1 - lambda2) / 2.0) <= -500000.0)
		tmp = Float64(Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))) * 2.0) * R);
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision], -500000.0], N[(N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < -5e5

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 31.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 21.9%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.3%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}} \cdot 2\right) \cdot R} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + \color{blue}{\frac{-1}{2} \cdot {\phi_1}^{2}}, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}} \cdot 2\right) \cdot R \]
   17. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot \color{blue}{{\phi_1}^{2}}, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{\color{blue}{2}}, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       3. lower-pow.f64 19.3%

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   18. Applied rewrites 19.3%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + \color{blue}{-0.5 \cdot {\phi_1}^{2}}, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   19. Taylor expanded in phi1 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + \color{blue}{\frac{-1}{2} \cdot {\phi_1}^{2}}, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}} \cdot 2\right) \cdot R \]
   20. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot \color{blue}{{\phi_1}^{2}}, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{\color{blue}{2}}, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}} \cdot 2\right) \cdot R \]

       3. lower-pow.f64 19.3%

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   21. Applied rewrites 19.3%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + \color{blue}{-0.5 \cdot {\phi_1}^{2}}, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}} \cdot 2\right) \cdot R \]

   if -5e5 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \phi_2\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-PI.f64 52.6%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\pi}, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 52.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \phi_2\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-PI.f64 51.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\pi}, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 51.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left({\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \color{blue}{\sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \color{blue}{\left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\color{blue}{\phi_2} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 39.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left({\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \color{blue}{\sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \color{blue}{\left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\color{blue}{\phi_2} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-PI.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower-pow.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 40.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.0%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 29.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.7%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 29.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 32: 29.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))
   (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (phi1 - phi2))), 2.0);
	return R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = sin((0.5d0 * (phi1 - phi2))) ** 2.0d0
    code = r * (2.0d0 * atan2(sqrt(t_0), sqrt((1.0d0 - t_0))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.pow(Math.sin((0.5 * (phi1 - phi2))), 2.0);
	return R * (2.0 * Math.atan2(Math.sqrt(t_0), Math.sqrt((1.0 - t_0))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin((0.5 * (phi1 - phi2))), 2.0)
	return R * (2.0 * math.atan2(math.sqrt(t_0), math.sqrt((1.0 - t_0))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0
	return Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin((0.5 * (phi1 - phi2))) ^ 2.0;
	tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. sin-+PI/2-rev N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. +-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \phi_2\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. metadata-eval N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-PI.f64 52.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\pi}, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 52.6%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
4. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. sin-+PI/2-rev N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\phi_2 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. +-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \phi_2\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. metadata-eval N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \phi_2\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-PI.f64 51.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\color{blue}{\pi}, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Applied rewrites 51.5%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

6. Taylor expanded in lambda2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left({\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
7. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \color{blue}{\sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \color{blue}{\left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\color{blue}{\phi_2} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-+.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. lower-PI.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, \frac{1}{2}, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

8. Applied rewrites 39.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, \phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

9. Taylor expanded in lambda2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left({\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right)\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
10. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    2. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2}} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \color{blue}{\sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    4. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \color{blue}{\left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    5. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\color{blue}{\phi_2} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    6. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    7. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    8. lower-+.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    9. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    10. lower-PI.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    11. lower-pow.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

11. Applied rewrites 40.1%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

12. Taylor expanded in lambda1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
13. Step-by-step derivation

    1. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    2. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + \frac{1}{2} \cdot \pi\right), {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    4. lower--.f64 29.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

14. Applied rewrites 29.0%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \lambda_1\right)}^{2} \cdot \sin \left(\phi_2 + 0.5 \cdot \pi\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

15. Taylor expanded in lambda1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
16. Step-by-step derivation

    1. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

    2. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

    4. lower--.f64 29.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

17. Applied rewrites 29.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2025188 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* R (* 2.0 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))) (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))))))))