Distance on a great circle

Percentage Accurate: 62.7% → 98.8%

Time: 38.5s

Alternatives: 36

Speedup: 1.4×

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: 62.7% 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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \cos \left(\lambda_2 \cdot 0.5\right) \cdot \sin \left(\lambda_1 \cdot 0.5\right)\\ t_2 := \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(-0.5 \cdot \phi_1\right)\\ t_3 := \sin \left(\lambda_2 \cdot -0.5\right) \cdot \cos \left(\lambda_1 \cdot -0.5\right)\\ t_4 := \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\\ t_5 := \left(\left({t\_4}^{2} - 2 \cdot \left(t\_4 \cdot t\_2\right)\right) + {t\_2}^{2}\right) + \mathsf{fma}\left({t\_1}^{2} - -2 \cdot \left(t\_1 \cdot t\_3\right), t\_0, {t\_3}^{2} \cdot t\_0\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (* (cos (* lambda2 0.5)) (sin (* lambda1 0.5))))
        (t_2 (* (sin (* phi2 0.5)) (cos (* -0.5 phi1))))
        (t_3 (* (sin (* lambda2 -0.5)) (cos (* lambda1 -0.5))))
        (t_4 (* (cos (* -0.5 phi2)) (sin (* phi1 0.5))))
        (t_5
         (+
          (+ (- (pow t_4 2.0) (* 2.0 (* t_4 t_2))) (pow t_2 2.0))
          (fma
           (- (pow t_1 2.0) (* -2.0 (* t_1 t_3)))
           t_0
           (* (pow t_3 2.0) t_0)))))
   (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = cos((lambda2 * 0.5)) * sin((lambda1 * 0.5));
	double t_2 = sin((phi2 * 0.5)) * cos((-0.5 * phi1));
	double t_3 = sin((lambda2 * -0.5)) * cos((lambda1 * -0.5));
	double t_4 = cos((-0.5 * phi2)) * sin((phi1 * 0.5));
	double t_5 = ((pow(t_4, 2.0) - (2.0 * (t_4 * t_2))) + pow(t_2, 2.0)) + fma((pow(t_1, 2.0) - (-2.0 * (t_1 * t_3))), t_0, (pow(t_3, 2.0) * t_0));
	return R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = Float64(cos(Float64(lambda2 * 0.5)) * sin(Float64(lambda1 * 0.5)))
	t_2 = Float64(sin(Float64(phi2 * 0.5)) * cos(Float64(-0.5 * phi1)))
	t_3 = Float64(sin(Float64(lambda2 * -0.5)) * cos(Float64(lambda1 * -0.5)))
	t_4 = Float64(cos(Float64(-0.5 * phi2)) * sin(Float64(phi1 * 0.5)))
	t_5 = Float64(Float64(Float64((t_4 ^ 2.0) - Float64(2.0 * Float64(t_4 * t_2))) + (t_2 ^ 2.0)) + fma(Float64((t_1 ^ 2.0) - Float64(-2.0 * Float64(t_1 * t_3))), t_0, Float64((t_3 ^ 2.0) * t_0)))
	return Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))))
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[N[(lambda2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(lambda1 * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Power[t$95$4, 2.0], $MachinePrecision] - N[(2.0 * N[(t$95$4 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] - N[(-2.0 * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[Power[t$95$3, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, 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]]]]]]]

Derivation

1. Initial program 62.7%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
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 61.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
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 63.2%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
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 63.0%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
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 77.5%

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

11. Applied rewrites 78.5%

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

13. Applied rewrites 98.8%

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

    1. lift-pow.f64 N/A

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

    2. lift--.f64 N/A

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

    3. sub-square-pow N/A

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

    4. lower-+.f64 N/A

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

15. Applied rewrites 98.8%

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

    1. lift-pow.f64 N/A

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

    2. lift--.f64 N/A

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

    3. sub-square-pow N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\left({\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)}^{2} - 2 \cdot \left(\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)\right)\right) + {\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)}^{2}\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_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(\left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} - 2 \cdot \left(\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\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. lower-+.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\left({\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)}^{2} - 2 \cdot \left(\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)\right)\right) + {\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)}^{2}\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_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(\left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} - 2 \cdot \left(\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\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. Applied rewrites 98.8%

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

19. Applied rewrites 98.7%

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

21. Applied rewrites 98.7%

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

Alternative 2: 98.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (* phi2 0.5)) (cos (* -0.5 phi1))))
        (t_1 (* (cos (* -0.5 phi2)) (sin (* phi1 0.5))))
        (t_2
         (+
          (+ (- (pow t_1 2.0) (* 2.0 (* t_1 t_0))) (pow t_0 2.0))
          (*
           (cos phi1)
           (*
            (cos phi2)
            (pow
             (fma
              (cos (* -0.5 lambda2))
              (sin (* 0.5 lambda1))
              (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
             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((phi2 * 0.5)) * cos((-0.5 * phi1));
	double t_1 = cos((-0.5 * phi2)) * sin((phi1 * 0.5));
	double t_2 = ((pow(t_1, 2.0) - (2.0 * (t_1 * t_0))) + pow(t_0, 2.0)) + (cos(phi1) * (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 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(phi2 * 0.5)) * cos(Float64(-0.5 * phi1)))
	t_1 = Float64(cos(Float64(-0.5 * phi2)) * sin(Float64(phi1 * 0.5)))
	t_2 = Float64(Float64(Float64((t_1 ^ 2.0) - Float64(2.0 * Float64(t_1 * t_0))) + (t_0 ^ 2.0)) + Float64(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))))
	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[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] - N[(2.0 * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + 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]), $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 62.7%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
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 61.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
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 63.2%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
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 63.0%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
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 77.5%

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

11. Applied rewrites 78.5%

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

13. Applied rewrites 98.8%

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

    1. lift-pow.f64 N/A

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

    2. lift--.f64 N/A

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

    3. sub-square-pow N/A

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

    4. lower-+.f64 N/A

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

15. Applied rewrites 98.8%

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

    1. lift-pow.f64 N/A

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

    2. lift--.f64 N/A

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

    3. sub-square-pow N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\left({\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)}^{2} - 2 \cdot \left(\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)\right)\right) + {\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)}^{2}\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_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(\left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} - 2 \cdot \left(\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\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. lower-+.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\left({\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)}^{2} - 2 \cdot \left(\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)\right)\right) + {\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)}^{2}\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_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(\left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} - 2 \cdot \left(\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\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. Applied rewrites 98.8%

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

18. Taylor expanded in lambda1 around inf

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-cos.f64 N/A

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

    5. lower-pow.f64 N/A

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

20. Applied rewrites 98.8%

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

21. Taylor expanded in lambda1 around inf

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-cos.f64 N/A

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

    5. lower-pow.f64 N/A

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

23. Applied rewrites 98.8%

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

Alternative 3: 98.7% 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 62.7%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
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 61.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
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 63.2%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
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 63.0%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
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 77.5%

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

11. Applied rewrites 78.5%

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

13. Applied rewrites 98.8%

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

14. Taylor expanded in lambda1 around inf

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

    1. lower-fma.f64 N/A

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

16. Applied rewrites 98.8%

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

17. Taylor expanded in lambda1 around inf

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
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 98.8%

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

Alternative 4: 87.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(-0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \sin \left(\phi_1 \cdot 0.5\right)\\ t_3 := \cos \left(\phi_1 \cdot 0.5\right)\\ t_4 := \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ t_5 := {\left(\mathsf{fma}\left(t\_2, \cos \left(\frac{\phi_2}{-2}\right), t\_3 \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + t\_4\\ t_6 := {\left(t\_2 \cdot \cos \left(\phi_2 \cdot 0.5\right) - t\_3 \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + t\_4\\ \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{+17}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_6}}{\sqrt{1 - t\_6}}\right)\\ \mathbf{elif}\;\phi_1 \leq 3.1 \cdot 10^{-56}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          (pow
           (-
            (* (sin (* 0.5 phi1)) (cos (* 0.5 phi2)))
            (* (cos (* -0.5 phi1)) (sin (* 0.5 phi2))))
           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_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (sin (* phi1 0.5)))
        (t_3 (cos (* phi1 0.5)))
        (t_4 (* (* (* (cos phi1) (cos phi2)) t_1) t_1))
        (t_5
         (+
          (pow (fma t_2 (cos (/ phi2 -2.0)) (* t_3 (sin (/ phi2 -2.0)))) 2.0)
          t_4))
        (t_6
         (+
          (pow (- (* t_2 (cos (* phi2 0.5))) (* t_3 (sin (* phi2 0.5)))) 2.0)
          t_4)))
   (if (<= phi1 -9e+17)
     (* R (* 2.0 (atan2 (sqrt t_6) (sqrt (- 1.0 t_6)))))
     (if (<= phi1 3.1e-56)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(((sin((0.5 * phi1)) * cos((0.5 * phi2))) - (cos((-0.5 * phi1)) * sin((0.5 * phi2)))), 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_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = sin((phi1 * 0.5));
	double t_3 = cos((phi1 * 0.5));
	double t_4 = ((cos(phi1) * cos(phi2)) * t_1) * t_1;
	double t_5 = pow(fma(t_2, cos((phi2 / -2.0)), (t_3 * sin((phi2 / -2.0)))), 2.0) + t_4;
	double t_6 = pow(((t_2 * cos((phi2 * 0.5))) - (t_3 * sin((phi2 * 0.5)))), 2.0) + t_4;
	double tmp;
	if (phi1 <= -9e+17) {
		tmp = R * (2.0 * atan2(sqrt(t_6), sqrt((1.0 - t_6))));
	} else if (phi1 <= 3.1e-56) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64((Float64(Float64(sin(Float64(0.5 * phi1)) * cos(Float64(0.5 * phi2))) - Float64(cos(Float64(-0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 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_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(phi1 * 0.5))
	t_3 = cos(Float64(phi1 * 0.5))
	t_4 = Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1)
	t_5 = Float64((fma(t_2, cos(Float64(phi2 / -2.0)), Float64(t_3 * sin(Float64(phi2 / -2.0)))) ^ 2.0) + t_4)
	t_6 = Float64((Float64(Float64(t_2 * cos(Float64(phi2 * 0.5))) - Float64(t_3 * sin(Float64(phi2 * 0.5)))) ^ 2.0) + t_4)
	tmp = 0.0
	if (phi1 <= -9e+17)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_6), sqrt(Float64(1.0 - t_6)))));
	elseif (phi1 <= 3.1e-56)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Power[N[(N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $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] + 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$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(t$95$2 * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(t$95$3 * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[Power[N[(N[(t$95$2 * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$4), $MachinePrecision]}, If[LessEqual[phi1, -9e+17], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$6], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$6), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 3.1e-56], 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], 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]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -9e17

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.6

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

   3. Applied rewrites 63.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 78.6

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

   5. Applied rewrites 78.6%

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

   if -9e17 < phi1 < 3.09999999999999987e-56

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   11. Applied rewrites 78.5%

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

   13. Applied rewrites 98.8%

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

   14. Taylor expanded in phi1 around 0

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

       1. lower-*.f64 N/A

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

       2. lower-cos.f64 N/A

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

       3. lower-pow.f64 N/A

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

   16. Applied rewrites 69.5%

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

   17. Taylor expanded in phi1 around 0

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

       1. lower-*.f64 N/A

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

       2. lower-cos.f64 N/A

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

       3. lower-pow.f64 N/A

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

   19. Applied rewrites 66.2%

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

   if 3.09999999999999987e-56 < phi1

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   3. Applied rewrites 63.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   5. Applied rewrites 78.6%

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

Alternative 5: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \sin \left(\phi_1 \cdot 0.5\right)\\ t_3 := \cos \left(\phi_1 \cdot 0.5\right)\\ t_4 := \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ t_5 := {\left(\mathsf{fma}\left(t\_2, \cos \left(\frac{\phi_2}{-2}\right), t\_3 \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + t\_4\\ t_6 := {\left(t\_2 \cdot \cos \left(\phi_2 \cdot 0.5\right) - t\_3 \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + t\_4\\ \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{+21}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_6}}{\sqrt{1 - t\_6}}\right)\\ \mathbf{elif}\;\phi_1 \leq 3.1 \cdot 10^{-56}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \end{array} \]

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 (sin (* 0.5 (- phi1 phi2))) 2.0)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (sin (* phi1 0.5)))
        (t_3 (cos (* phi1 0.5)))
        (t_4 (* (* (* (cos phi1) (cos phi2)) t_1) t_1))
        (t_5
         (+
          (pow (fma t_2 (cos (/ phi2 -2.0)) (* t_3 (sin (/ phi2 -2.0)))) 2.0)
          t_4))
        (t_6
         (+
          (pow (- (* t_2 (cos (* phi2 0.5))) (* t_3 (sin (* phi2 0.5)))) 2.0)
          t_4)))
   (if (<= phi1 -1.1e+21)
     (* R (* 2.0 (atan2 (sqrt t_6) (sqrt (- 1.0 t_6)))))
     (if (<= phi1 3.1e-56)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))))))

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(sin((0.5 * (phi1 - phi2))), 2.0));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = sin((phi1 * 0.5));
	double t_3 = cos((phi1 * 0.5));
	double t_4 = ((cos(phi1) * cos(phi2)) * t_1) * t_1;
	double t_5 = pow(fma(t_2, cos((phi2 / -2.0)), (t_3 * sin((phi2 / -2.0)))), 2.0) + t_4;
	double t_6 = pow(((t_2 * cos((phi2 * 0.5))) - (t_3 * sin((phi2 * 0.5)))), 2.0) + t_4;
	double tmp;
	if (phi1 <= -1.1e+21) {
		tmp = R * (2.0 * atan2(sqrt(t_6), sqrt((1.0 - t_6))));
	} else if (phi1 <= 3.1e-56) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	}
	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)), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(phi1 * 0.5))
	t_3 = cos(Float64(phi1 * 0.5))
	t_4 = Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1)
	t_5 = Float64((fma(t_2, cos(Float64(phi2 / -2.0)), Float64(t_3 * sin(Float64(phi2 / -2.0)))) ^ 2.0) + t_4)
	t_6 = Float64((Float64(Float64(t_2 * cos(Float64(phi2 * 0.5))) - Float64(t_3 * sin(Float64(phi2 * 0.5)))) ^ 2.0) + t_4)
	tmp = 0.0
	if (phi1 <= -1.1e+21)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_6), sqrt(Float64(1.0 - t_6)))));
	elseif (phi1 <= 3.1e-56)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	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[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(t$95$2 * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(t$95$3 * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[Power[N[(N[(t$95$2 * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$4), $MachinePrecision]}, If[LessEqual[phi1, -1.1e+21], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$6], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$6), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 3.1e-56], 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], 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]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.1e21

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.6

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

   3. Applied rewrites 63.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 78.6

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

   5. Applied rewrites 78.6%

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

   if -1.1e21 < phi1 < 3.09999999999999987e-56

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in lambda1 around inf

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

       1. lower-fma.f64 N/A

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

   12. Applied rewrites 77.6%

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

   13. Taylor expanded in lambda1 around inf

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

       1. lower-fma.f64 N/A

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

   15. Applied rewrites 77.6%

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

   if 3.09999999999999987e-56 < phi1

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   3. Applied rewrites 63.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   5. Applied rewrites 78.6%

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

Alternative 6: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{+21}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_1 \leq 3.1 \cdot 10^{-56}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

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 (sin (* 0.5 (- phi1 phi2))) 2.0)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (+
          (pow
           (-
            (* (sin (* phi1 0.5)) (cos (* phi2 0.5)))
            (* (cos (* phi1 0.5)) (sin (* phi2 0.5))))
           2.0)
          (* (* (* (cos phi1) (cos phi2)) t_1) t_1)))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi1 -1.1e+21)
     t_3
     (if (<= phi1 3.1e-56)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_3))))

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(sin((0.5 * (phi1 - phi2))), 2.0));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(((sin((phi1 * 0.5)) * cos((phi2 * 0.5))) - (cos((phi1 * 0.5)) * sin((phi2 * 0.5)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1);
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi1 <= -1.1e+21) {
		tmp = t_3;
	} else if (phi1 <= 3.1e-56) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_3;
	}
	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)), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = 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_1) * t_1))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi1 <= -1.1e+21)
		tmp = t_3;
	elseif (phi1 <= 3.1e-56)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_3;
	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[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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$1), $MachinePrecision] * t$95$1), $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[phi1, -1.1e+21], t$95$3, If[LessEqual[phi1, 3.1e-56], 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$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.1e21 or 3.09999999999999987e-56 < phi1

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.6

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

   3. Applied rewrites 63.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 78.6

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

   5. Applied rewrites 78.6%

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

   if -1.1e21 < phi1 < 3.09999999999999987e-56

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in lambda1 around inf

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

       1. lower-fma.f64 N/A

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

   12. Applied rewrites 77.6%

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

   13. Taylor expanded in lambda1 around inf

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

       1. lower-fma.f64 N/A

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

   15. Applied rewrites 77.6%

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

Alternative 7: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)\\ t_1 := {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(-0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{+21}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 1.8 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (*
           (cos phi2)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
            2.0))
          (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))
        (t_1
         (+
          (pow
           (-
            (* (sin (* 0.5 phi1)) (cos (* 0.5 phi2)))
            (* (cos (* -0.5 phi1)) (sin (* 0.5 phi2))))
           2.0)
          (*
           (* (cos phi2) (cos phi1))
           (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5))))))))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= phi1 -1.1e+21)
     t_2
     (if (<= phi1 1.8e-10)
       (* 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) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0)), pow(sin((0.5 * (phi1 - phi2))), 2.0));
	double t_1 = pow(((sin((0.5 * phi1)) * cos((0.5 * phi2))) - (cos((-0.5 * phi1)) * sin((0.5 * phi2)))), 2.0) + ((cos(phi2) * cos(phi1)) * (0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))));
	double t_2 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double tmp;
	if (phi1 <= -1.1e+21) {
		tmp = t_2;
	} else if (phi1 <= 1.8e-10) {
		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) * (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 * Float64(phi1 - phi2))) ^ 2.0))
	t_1 = Float64((Float64(Float64(sin(Float64(0.5 * phi1)) * cos(Float64(0.5 * phi2))) - Float64(cos(Float64(-0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0) + Float64(Float64(cos(phi2) * cos(phi1)) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5)))))))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	tmp = 0.0
	if (phi1 <= -1.1e+21)
		tmp = t_2;
	elseif (phi1 <= 1.8e-10)
		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[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[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $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] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $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[phi1, -1.1e+21], t$95$2, If[LessEqual[phi1, 1.8e-10], 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 phi1 < -1.1e21 or 1.8e-10 < phi1

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   11. Applied rewrites 78.5%

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

   13. Applied rewrites 98.8%

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

   15. Applied rewrites 76.6%

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

   17. Applied rewrites 76.0%

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

   if -1.1e21 < phi1 < 1.8e-10

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in lambda1 around inf

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

       1. lower-fma.f64 N/A

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

   12. Applied rewrites 77.6%

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

   13. Taylor expanded in lambda1 around inf

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

       1. lower-fma.f64 N/A

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

   15. Applied rewrites 77.6%

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

Alternative 8: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(-0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{+21}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 1.8 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (*
           (cos phi2)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
            2.0))))
        (t_1
         (+
          (pow
           (-
            (* (sin (* 0.5 phi1)) (cos (* 0.5 phi2)))
            (* (cos (* -0.5 phi1)) (sin (* 0.5 phi2))))
           2.0)
          (*
           (* (cos phi2) (cos phi1))
           (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5))))))))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= phi1 -1.1e+21)
     t_2
     (if (<= phi1 1.8e-10)
       (* 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(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0));
	double t_1 = pow(((sin((0.5 * phi1)) * cos((0.5 * phi2))) - (cos((-0.5 * phi1)) * sin((0.5 * phi2)))), 2.0) + ((cos(phi2) * cos(phi1)) * (0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))));
	double t_2 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double tmp;
	if (phi1 <= -1.1e+21) {
		tmp = t_2;
	} else if (phi1 <= 1.8e-10) {
		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(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_1 = Float64((Float64(Float64(sin(Float64(0.5 * phi1)) * cos(Float64(0.5 * phi2))) - Float64(cos(Float64(-0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0) + Float64(Float64(cos(phi2) * cos(phi1)) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5)))))))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	tmp = 0.0
	if (phi1 <= -1.1e+21)
		tmp = t_2;
	elseif (phi1 <= 1.8e-10)
		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[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$1 = N[(N[Power[N[(N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $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] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $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[phi1, -1.1e+21], t$95$2, If[LessEqual[phi1, 1.8e-10], 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 phi1 < -1.1e21 or 1.8e-10 < phi1

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   11. Applied rewrites 78.5%

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

   13. Applied rewrites 98.8%

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

   15. Applied rewrites 76.6%

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

   17. Applied rewrites 76.0%

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

   if -1.1e21 < phi1 < 1.8e-10

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. 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 63.3%

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

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   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 61.1%

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

Alternative 9: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot t\_0\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -0.018:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 0.24:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

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 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi1) t_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 -0.018)
     t_3
     (if (<= phi2 0.24)
       (* 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 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * t_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 <= -0.018) {
		tmp = t_3;
	} else if (phi2 <= 0.24) {
		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 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * t_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 <= -0.018)
		tmp = t_3;
	elseif (phi2 <= 0.24)
		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[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$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, -0.018], t$95$3, If[LessEqual[phi2, 0.24], 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 < -0.0179999999999999986 or 0.23999999999999999 < phi2

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. 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 56.2%

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

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   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 56.2%

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

   if -0.0179999999999999986 < phi2 < 0.23999999999999999

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. 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

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   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 10: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\phi_1 \leq -215000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 2.85 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot t\_0}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

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 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= phi1 -215000000000.0)
     t_2
     (if (<= phi1 2.85e-5)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi2) t_0)))
          (sqrt (- 1.0 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))))))
       t_2))))

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 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double tmp;
	if (phi1 <= -215000000000.0) {
		tmp = t_2;
	} else if (phi1 <= 2.85e-5) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi2) * t_0))), sqrt((1.0 - fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0))))));
	} else {
		tmp = t_2;
	}
	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 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	tmp = 0.0
	if (phi1 <= -215000000000.0)
		tmp = t_2;
	elseif (phi1 <= 2.85e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi2) * t_0))), sqrt(Float64(1.0 - fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0)))))));
	else
		tmp = t_2;
	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[(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[phi1, -215000000000.0], t$95$2, If[LessEqual[phi1, 2.85e-5], 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[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.15e11 or 2.8500000000000002e-5 < phi1

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. 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 57.5%

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

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 57.5%

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

   if -2.15e11 < phi1 < 2.8500000000000002e-5

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. 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 63.3%

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

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   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 61.1%

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

   16. Taylor expanded in phi1 around 0

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

       1. lower--.f64 N/A

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

       2. lower-fma.f64 N/A

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

   18. Applied rewrites 58.5%

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

Alternative 11: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_1 \leq -215000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_1 \leq 1.3 \cdot 10^{-30}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

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 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 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi1 -215000000000.0)
     t_3
     (if (<= phi1 1.3e-30)
       (* 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(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 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi1 <= -215000000000.0) {
		tmp = t_3;
	} else if (phi1 <= 1.3e-30) {
		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(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 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi1 <= -215000000000.0)
		tmp = t_3;
	elseif (phi1 <= 1.3e-30)
		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[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[(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[phi1, -215000000000.0], t$95$3, If[LessEqual[phi1, 1.3e-30], 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 phi1 < -2.15e11 or 1.29999999999999993e-30 < phi1

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. 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 57.5%

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

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 57.5%

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

   if -2.15e11 < phi1 < 1.29999999999999993e-30

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. 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 56.2%

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

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   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 56.2%

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

Alternative 12: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {t\_0}^{2}\right)\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\\ t_5 := \frac{\left(\cos \left(\frac{\left(\phi_2 - \phi_1\right) - \left(\phi_2 - \phi_1\right)}{-2}\right) - t\_4\right) + \left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot t\_1}{2}\\ \mathbf{if}\;\phi_2 \leq -0.082:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{elif}\;\phi_2 \leq 0.24:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(t\_0, \cos \left(0.5 \cdot \phi_2\right), \left(-\sin \left(0.5 \cdot \phi_2\right)\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_3\right) \cdot t\_3}}{e^{\log \left(\left(0.5 + 0.5 \cdot t\_4\right) - t\_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right) \cdot 0.5}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
        (t_2
         (fma
          (cos phi1)
          (pow
           (fma
            (cos (* -0.5 lambda2))
            (sin (* 0.5 lambda1))
            (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
           2.0)
          (pow t_0 2.0)))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_4 (cos (* 2.0 (* 0.5 (- phi1 phi2)))))
        (t_5
         (/
          (+
           (- (cos (/ (- (- phi2 phi1) (- phi2 phi1)) -2.0)) t_4)
           (* (+ (cos (- phi2 phi1)) (cos (+ phi2 phi1))) t_1))
          2.0)))
   (if (<= phi2 -0.082)
     (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
     (if (<= phi2 0.24)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (+
            (pow
             (fma
              t_0
              (cos (* 0.5 phi2))
              (* (- (sin (* 0.5 phi2))) (cos (* 0.5 phi1))))
             2.0)
            (* (* (* (cos phi1) (cos phi2)) t_3) t_3)))
          (exp
           (*
            (log (- (+ 0.5 (* 0.5 t_4)) (* t_1 (* (cos phi2) (cos phi1)))))
            0.5)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1));
	double t_1 = 0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))));
	double t_2 = fma(cos(phi1), pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0), pow(t_0, 2.0));
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double t_4 = cos((2.0 * (0.5 * (phi1 - phi2))));
	double t_5 = ((cos((((phi2 - phi1) - (phi2 - phi1)) / -2.0)) - t_4) + ((cos((phi2 - phi1)) + cos((phi2 + phi1))) * t_1)) / 2.0;
	double tmp;
	if (phi2 <= -0.082) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else if (phi2 <= 0.24) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(fma(t_0, cos((0.5 * phi2)), (-sin((0.5 * phi2)) * cos((0.5 * phi1)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_3) * t_3))), exp((log(((0.5 + (0.5 * t_4)) - (t_1 * (cos(phi2) * cos(phi1))))) * 0.5))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi1))
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))
	t_2 = fma(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_0 ^ 2.0))
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_4 = cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2))))
	t_5 = Float64(Float64(Float64(cos(Float64(Float64(Float64(phi2 - phi1) - Float64(phi2 - phi1)) / -2.0)) - t_4) + Float64(Float64(cos(Float64(phi2 - phi1)) + cos(Float64(phi2 + phi1))) * t_1)) / 2.0)
	tmp = 0.0
	if (phi2 <= -0.082)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	elseif (phi2 <= 0.24)
		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((fma(t_0, cos(Float64(0.5 * phi2)), Float64(Float64(-sin(Float64(0.5 * phi2))) * cos(Float64(0.5 * phi1)))) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_3) * t_3))), exp(Float64(log(Float64(Float64(0.5 + Float64(0.5 * t_4)) - Float64(t_1 * Float64(cos(phi2) * cos(phi1))))) * 0.5)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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$2 = 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] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Cos[N[(N[(N[(phi2 - phi1), $MachinePrecision] - N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]], $MachinePrecision] - t$95$4), $MachinePrecision] + N[(N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[phi2, -0.082], 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], If[LessEqual[phi2, 0.24], 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[(t$95$0 * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] + N[((-N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]) * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[Log[N[(N[(0.5 + N[(0.5 * t$95$4), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.0820000000000000034

   1. Initial program 62.7%

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

   3. Applied rewrites 58.2%

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

   5. Applied rewrites 58.6%

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

   if -0.0820000000000000034 < phi2 < 0.23999999999999999

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. 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 57.5%

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

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 57.5%

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

   if 0.23999999999999999 < phi2

   1. Initial program 62.7%

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

   3. Applied rewrites 62.8%

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

   5. Applied rewrites 63.7%

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

Alternative 13: 63.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot t\_0\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -0.082:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 0.021:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi1) t_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 -0.082)
     t_3
     (if (<= phi2 0.021)
       (* 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(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * t_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 <= -0.082) {
		tmp = t_3;
	} else if (phi2 <= 0.021) {
		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 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * t_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 <= -0.082)
		tmp = t_3;
	elseif (phi2 <= 0.021)
		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[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[phi1], $MachinePrecision] * t$95$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, -0.082], t$95$3, If[LessEqual[phi2, 0.021], 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 < -0.0820000000000000034 or 0.0210000000000000013 < phi2

   1. Initial program 62.7%

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

   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 46.3

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

   4. Applied rewrites 46.3%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   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 46.5

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

   7. Applied rewrites 46.5%

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

   if -0.0820000000000000034 < phi2 < 0.0210000000000000013

   1. Initial program 62.7%

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

   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 53.5

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   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 51.4

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

   7. Applied rewrites 51.4%

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

Alternative 14: 63.7% accurate, 0.8× speedup

Math

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

FPCore

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

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(fma(sin((0.5 * phi1)), cos((0.5 * phi2)), (-sin((0.5 * phi2)) * cos((0.5 * phi1)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), exp((log(((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))))) * 0.5))));
}

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((fma(sin(Float64(0.5 * phi1)), cos(Float64(0.5 * phi2)), Float64(Float64(-sin(Float64(0.5 * phi2))) * cos(Float64(0.5 * phi1)))) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))), exp(Float64(log(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))))) * 0.5)))))
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[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] + N[((-N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]) * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $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[Exp[N[(N[Log[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] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.7%

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

3. Applied rewrites 62.8%

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

5. Applied rewrites 63.7%

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

Alternative 15: 63.5% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow
         (-
          (* (sin (* 0.5 phi1)) (cos (* 0.5 phi2)))
          (* (cos (* 0.5 phi1)) (sin (* 0.5 phi2))))
         2.0)
        (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
      (exp
       (*
        (log
         (-
          (+ 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)))))
        0.5)))))))

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((0.5 * phi1)) * cos((0.5 * phi2))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), exp((log(((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))))) * 0.5))));
}

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((0.5d0 * phi1)) * cos((0.5d0 * phi2))) - (cos((0.5d0 * phi1)) * sin((0.5d0 * phi2)))) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), exp((log(((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))))) * 0.5d0))))
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((0.5 * phi1)) * Math.cos((0.5 * phi2))) - (Math.cos((0.5 * phi1)) * Math.sin((0.5 * phi2)))), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * t_0) * t_0))), Math.exp((Math.log(((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))))) * 0.5))));
}

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((0.5 * phi1)) * math.cos((0.5 * phi2))) - (math.cos((0.5 * phi1)) * math.sin((0.5 * phi2)))), 2.0) + (((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0))), math.exp((math.log(((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))))) * 0.5))))

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((Float64(Float64(sin(Float64(0.5 * phi1)) * cos(Float64(0.5 * phi2))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))), exp(Float64(log(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))))) * 0.5)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	tmp = R * (2.0 * atan2(sqrt(((((sin((0.5 * phi1)) * cos((0.5 * phi2))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))) ^ 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), exp((log(((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))))) * 0.5))));
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[(N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $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] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[Log[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] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.7%

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

3. Applied rewrites 62.8%

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

5. Applied rewrites 63.7%

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

Alternative 16: 63.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\\ t_2 := 0.5 \cdot \left(\lambda_1 - \lambda_2\right)\\ t_3 := 0.5 - 0.5 \cdot \cos \left(2 \cdot t\_2\right)\\ t_4 := \cos \phi_2 \cdot \cos \phi_1\\ t_5 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_6 := \left(0.5 + t\_1\right) - t\_3 \cdot t\_4\\ \mathbf{if}\;t\_5 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0 \leq 0.002:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5 + \cos \phi_1 \cdot {\sin t\_2}^{2}}}{e^{\log t\_6 \cdot 0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_3, t\_4, 0.5 - t\_1\right)}}{\sqrt{t\_6}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
        (t_2 (* 0.5 (- lambda1 lambda2)))
        (t_3 (- 0.5 (* 0.5 (cos (* 2.0 t_2)))))
        (t_4 (* (cos phi2) (cos phi1)))
        (t_5 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_6 (- (+ 0.5 t_1) (* t_3 t_4))))
   (if (<= (+ t_5 (* (* (* (cos phi1) (cos phi2)) t_0) t_0)) 0.002)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_5 (* (cos phi1) (pow (sin t_2) 2.0))))
        (exp (* (log t_6) 0.5)))))
     (* (* (atan2 (sqrt (fma t_3 t_4 (- 0.5 t_1))) (sqrt t_6)) 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 = 0.5 * cos((2.0 * (0.5 * (phi1 - phi2))));
	double t_2 = 0.5 * (lambda1 - lambda2);
	double t_3 = 0.5 - (0.5 * cos((2.0 * t_2)));
	double t_4 = cos(phi2) * cos(phi1);
	double t_5 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_6 = (0.5 + t_1) - (t_3 * t_4);
	double tmp;
	if ((t_5 + (((cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 0.002) {
		tmp = R * (2.0 * atan2(sqrt((t_5 + (cos(phi1) * pow(sin(t_2), 2.0)))), exp((log(t_6) * 0.5))));
	} else {
		tmp = (atan2(sqrt(fma(t_3, t_4, (0.5 - t_1))), sqrt(t_6)) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))
	t_2 = Float64(0.5 * Float64(lambda1 - lambda2))
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_2))))
	t_4 = Float64(cos(phi2) * cos(phi1))
	t_5 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_6 = Float64(Float64(0.5 + t_1) - Float64(t_3 * t_4))
	tmp = 0.0
	if (Float64(t_5 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 0.002)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_5 + Float64(cos(phi1) * (sin(t_2) ^ 2.0)))), exp(Float64(log(t_6) * 0.5)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(t_3, t_4, Float64(0.5 - t_1))), sqrt(t_6)) * 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[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$6 = N[(N[(0.5 + t$95$1), $MachinePrecision] - N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$5 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 0.002], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$5 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[t$95$2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[Log[t$95$6], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$3 * t$95$4 + N[(0.5 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[t$95$6], $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))))) < 2e-3

   1. Initial program 62.7%

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

   3. Applied rewrites 62.8%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 53.6

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

   6. Applied rewrites 53.6%

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

   if 2e-3 < (+.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 62.7%

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

   3. Applied rewrites 57.8%

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

Alternative 17: 63.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := \sin \left(0.5 \cdot \phi_1\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_1, t\_0, \frac{1}{{t\_2}^{-2}}\right)\\ \mathbf{if}\;\phi_1 \leq -215000000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\frac{1}{{\left(\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\right)}^{-0.5}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_0, {t\_2}^{2}\right)}}\right)\\ \mathbf{elif}\;\phi_1 \leq 1.3 \cdot 10^{-30}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \end{array} \]

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 (sin (* 0.5 phi1)))
        (t_3 (fma (cos phi1) t_0 (/ 1.0 (pow t_2 -2.0)))))
   (if (<= phi1 -215000000000.0)
     (*
      R
      (*
       2.0
       (atan2
        (/
         1.0
         (pow
          (fma
           (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
           (cos phi1)
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1))))))
          -0.5))
        (sqrt (- 1.0 (fma (cos phi1) t_0 (pow t_2 2.0)))))))
     (if (<= phi1 1.3e-30)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))))))

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 = sin((0.5 * phi1));
	double t_3 = fma(cos(phi1), t_0, (1.0 / pow(t_2, -2.0)));
	double tmp;
	if (phi1 <= -215000000000.0) {
		tmp = R * (2.0 * atan2((1.0 / pow(fma((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1)))))), -0.5)), sqrt((1.0 - fma(cos(phi1), t_0, pow(t_2, 2.0))))));
	} else if (phi1 <= 1.3e-30) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	}
	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 = sin(Float64(0.5 * phi1))
	t_3 = fma(cos(phi1), t_0, Float64(1.0 / (t_2 ^ -2.0)))
	tmp = 0.0
	if (phi1 <= -215000000000.0)
		tmp = Float64(R * Float64(2.0 * atan(Float64(1.0 / (fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))) ^ -0.5)), sqrt(Float64(1.0 - fma(cos(phi1), t_0, (t_2 ^ 2.0)))))));
	elseif (phi1 <= 1.3e-30)
		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(t_3), sqrt(Float64(1.0 - t_3)))));
	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[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[(1.0 / N[Power[t$95$2, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -215000000000.0], N[(R * N[(2.0 * N[ArcTan[N[(1.0 / N[Power[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 1.3e-30], 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[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.15e11

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-unsound-/.f64 N/A

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

   9. Applied rewrites 43.4%

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

   if -2.15e11 < phi1 < 1.29999999999999993e-30

   1. Initial program 62.7%

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

   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 46.3

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

   4. Applied rewrites 46.3%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   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 46.5

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

   7. Applied rewrites 46.5%

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

   if 1.29999999999999993e-30 < phi1

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

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

      3. pow-neg N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. lower-unsound-pow.f64 47.5

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

   9. Applied rewrites 47.5%

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

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-neg N/A

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

       4. lower-unsound-/.f64 N/A

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

       5. lower-unsound-pow.f64 47.5

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

   11. Applied rewrites 47.5%

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

Alternative 18: 63.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-1 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := \cos \left(-1 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_4 := 0.5 \cdot \left(\lambda_1 - \lambda_2\right)\\ t_5 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;t\_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1 \leq 0.002:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2 + \cos \phi_1 \cdot {\sin t\_4}^{2}}}{e^{\log \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_4\right)\right) \cdot t\_5\right) \cdot 0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(0.5 - t\_3 \cdot 0.5\right) \cdot \cos \phi_2, \cos \phi_1, 0.5 - t\_0 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(t\_0 + 1, 0.5, \mathsf{fma}\left(t\_3, 0.5, -0.5\right) \cdot t\_5\right)}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* -1.0 (- phi1 phi2))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_3 (cos (* -1.0 (- lambda1 lambda2))))
        (t_4 (* 0.5 (- lambda1 lambda2)))
        (t_5 (* (cos phi2) (cos phi1))))
   (if (<= (+ t_2 (* (* (* (cos phi1) (cos phi2)) t_1) t_1)) 0.002)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_2 (* (cos phi1) (pow (sin t_4) 2.0))))
        (exp
         (*
          (log
           (-
            (+ 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
            (* (- 0.5 (* 0.5 (cos (* 2.0 t_4)))) t_5)))
          0.5)))))
     (*
      (*
       (atan2
        (sqrt
         (fma
          (* (- 0.5 (* t_3 0.5)) (cos phi2))
          (cos phi1)
          (- 0.5 (* t_0 0.5))))
        (sqrt (fma (+ t_0 1.0) 0.5 (* (fma t_3 0.5 -0.5) t_5))))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((-1.0 * (phi1 - phi2)));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_3 = cos((-1.0 * (lambda1 - lambda2)));
	double t_4 = 0.5 * (lambda1 - lambda2);
	double t_5 = cos(phi2) * cos(phi1);
	double tmp;
	if ((t_2 + (((cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.002) {
		tmp = R * (2.0 * atan2(sqrt((t_2 + (cos(phi1) * pow(sin(t_4), 2.0)))), exp((log(((0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * cos((2.0 * t_4)))) * t_5))) * 0.5))));
	} else {
		tmp = (atan2(sqrt(fma(((0.5 - (t_3 * 0.5)) * cos(phi2)), cos(phi1), (0.5 - (t_0 * 0.5)))), sqrt(fma((t_0 + 1.0), 0.5, (fma(t_3, 0.5, -0.5) * t_5)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(-1.0 * Float64(phi1 - phi2)))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_3 = cos(Float64(-1.0 * Float64(lambda1 - lambda2)))
	t_4 = Float64(0.5 * Float64(lambda1 - lambda2))
	t_5 = Float64(cos(phi2) * cos(phi1))
	tmp = 0.0
	if (Float64(t_2 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.002)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_2 + Float64(cos(phi1) * (sin(t_4) ^ 2.0)))), exp(Float64(log(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 * t_4)))) * t_5))) * 0.5)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(Float64(Float64(0.5 - Float64(t_3 * 0.5)) * cos(phi2)), cos(phi1), Float64(0.5 - Float64(t_0 * 0.5)))), sqrt(fma(Float64(t_0 + 1.0), 0.5, Float64(fma(t_3, 0.5, -0.5) * t_5)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(-1.0 * N[(phi1 - phi2), $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[Cos[N[(-1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], 0.002], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[t$95$4], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[Log[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 * t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[(0.5 - N[(t$95$3 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 + 1.0), $MachinePrecision] * 0.5 + N[(N[(t$95$3 * 0.5 + -0.5), $MachinePrecision] * t$95$5), $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))))) < 2e-3

   1. Initial program 62.7%

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

   3. Applied rewrites 62.8%

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

   4. Taylor expanded in phi2 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 53.6

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

   6. Applied rewrites 53.6%

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

   if 2e-3 < (+.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 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   11. Applied rewrites 78.5%

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

   13. Applied rewrites 98.8%

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

       1. lift-pow.f64 N/A

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

       2. lift--.f64 N/A

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

       3. sub-square-pow N/A

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

       4. lower-+.f64 N/A

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

   15. Applied rewrites 98.8%

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

       1. lift-pow.f64 N/A

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

       2. lift--.f64 N/A

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

       3. sub-square-pow N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\left({\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)}^{2} - 2 \cdot \left(\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)\right)\right) + {\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)}^{2}\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_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(\left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} - 2 \cdot \left(\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\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. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\left({\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)}^{2} - 2 \cdot \left(\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)\right)\right) + {\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)}^{2}\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_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(\left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} - 2 \cdot \left(\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\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. Applied rewrites 98.8%

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

   19. Applied rewrites 57.8%

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

Alternative 19: 62.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_2 := \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ \mathbf{if}\;\phi_1 \leq -215000000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\frac{1}{{t\_2}^{-0.5}}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{elif}\;\phi_1 \leq 1.3 \cdot 10^{-30}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
        (t_2
         (fma
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
          (cos phi1)
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
        (t_3 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0))))
   (if (<= phi1 -215000000000.0)
     (* R (* 2.0 (atan2 (/ 1.0 (pow t_2 -0.5)) (sqrt (- 1.0 t_1)))))
     (if (<= phi1 1.3e-30)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 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 = fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0));
	double t_2 = fma((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))));
	double t_3 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double tmp;
	if (phi1 <= -215000000000.0) {
		tmp = R * (2.0 * atan2((1.0 / pow(t_2, -0.5)), sqrt((1.0 - t_1))));
	} else if (phi1 <= 1.3e-30) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - 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 = fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_2 = fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	t_3 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	tmp = 0.0
	if (phi1 <= -215000000000.0)
		tmp = Float64(R * Float64(2.0 * atan(Float64(1.0 / (t_2 ^ -0.5)), sqrt(Float64(1.0 - t_1)))));
	elseif (phi1 <= 1.3e-30)
		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_1), sqrt(Float64(1.0 - 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[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[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -215000000000.0], N[(R * N[(2.0 * N[ArcTan[N[(1.0 / N[Power[t$95$2, -0.5], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 1.3e-30], 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$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.15e11

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-unsound-/.f64 N/A

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

   9. Applied rewrites 43.4%

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

   if -2.15e11 < phi1 < 1.29999999999999993e-30

   1. Initial program 62.7%

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

   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 46.3

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

   4. Applied rewrites 46.3%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   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 46.5

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

   7. Applied rewrites 46.5%

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

   if 1.29999999999999993e-30 < phi1

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 47.5

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

   9. Applied rewrites 47.5%

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

Alternative 20: 62.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-1 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \left(-1 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := t\_3 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;t\_3 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1 \leq 0.002:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(0.5 - t\_0 \cdot 0.5\right) \cdot \cos \phi_2, \cos \phi_1, 0.5 - t\_2 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(t\_2 + 1, 0.5, \mathsf{fma}\left(t\_0, 0.5, -0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* -1.0 (- lambda1 lambda2))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (cos (* -1.0 (- phi1 phi2))))
        (t_3 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_4
         (+ t_3 (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))))
   (if (<= (+ t_3 (* (* (* (cos phi1) (cos phi2)) t_1) t_1)) 0.002)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (*
      (*
       (atan2
        (sqrt
         (fma
          (* (- 0.5 (* t_0 0.5)) (cos phi2))
          (cos phi1)
          (- 0.5 (* t_2 0.5))))
        (sqrt
         (fma
          (+ t_2 1.0)
          0.5
          (* (fma t_0 0.5 -0.5) (* (cos phi2) (cos phi1))))))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((-1.0 * (lambda1 - lambda2)));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = cos((-1.0 * (phi1 - phi2)));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_4 = t_3 + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double tmp;
	if ((t_3 + (((cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.002) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else {
		tmp = (atan2(sqrt(fma(((0.5 - (t_0 * 0.5)) * cos(phi2)), cos(phi1), (0.5 - (t_2 * 0.5)))), sqrt(fma((t_2 + 1.0), 0.5, (fma(t_0, 0.5, -0.5) * (cos(phi2) * cos(phi1)))))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(-1.0 * Float64(lambda1 - lambda2)))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = cos(Float64(-1.0 * Float64(phi1 - phi2)))
	t_3 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_4 = 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_1) * t_1)) <= 0.002)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(Float64(Float64(0.5 - Float64(t_0 * 0.5)) * cos(phi2)), cos(phi1), Float64(0.5 - Float64(t_2 * 0.5)))), sqrt(fma(Float64(t_2 + 1.0), 0.5, Float64(fma(t_0, 0.5, -0.5) * Float64(cos(phi2) * cos(phi1)))))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(-1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(-1.0 * N[(phi1 - phi2), $MachinePrecision]), $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[(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$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 0.002], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(N[(N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$2 + 1.0), $MachinePrecision] * 0.5 + N[(N[(t$95$0 * 0.5 + -0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $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))))) < 2e-3

   1. Initial program 62.7%

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

   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 53.5

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   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 51.4

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

   7. Applied rewrites 51.4%

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

   if 2e-3 < (+.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 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   11. Applied rewrites 78.5%

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

   13. Applied rewrites 98.8%

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

       1. lift-pow.f64 N/A

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

       2. lift--.f64 N/A

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

       3. sub-square-pow N/A

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

       4. lower-+.f64 N/A

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

   15. Applied rewrites 98.8%

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

       1. lift-pow.f64 N/A

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

       2. lift--.f64 N/A

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

       3. sub-square-pow N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\left({\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)}^{2} - 2 \cdot \left(\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)\right)\right) + {\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)}^{2}\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_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(\left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} - 2 \cdot \left(\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\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. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\left({\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)}^{2} - 2 \cdot \left(\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)\right)\right) + {\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \phi_1\right)\right)}^{2}\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_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(\left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} - 2 \cdot \left(\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\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. Applied rewrites 98.8%

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

   19. Applied rewrites 57.8%

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

Alternative 21: 62.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_2 := \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ \mathbf{if}\;\phi_1 \leq -215000000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{elif}\;\phi_1 \leq 1.3 \cdot 10^{-30}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
        (t_2
         (fma
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
          (cos phi1)
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
        (t_3 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0))))
   (if (<= phi1 -215000000000.0)
     (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_1)))))
     (if (<= phi1 1.3e-30)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 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 = fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0));
	double t_2 = fma((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))));
	double t_3 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double tmp;
	if (phi1 <= -215000000000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_1))));
	} else if (phi1 <= 1.3e-30) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - 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 = fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_2 = fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	t_3 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	tmp = 0.0
	if (phi1 <= -215000000000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_1)))));
	elseif (phi1 <= 1.3e-30)
		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_1), sqrt(Float64(1.0 - 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[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[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -215000000000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 1.3e-30], 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$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.15e11

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 47.5

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

   9. Applied rewrites 43.4%

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

   if -2.15e11 < phi1 < 1.29999999999999993e-30

   1. Initial program 62.7%

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

   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 46.3

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

   4. Applied rewrites 46.3%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   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 46.5

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

   7. Applied rewrites 46.5%

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

   if 1.29999999999999993e-30 < phi1

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 47.5

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

   9. Applied rewrites 47.5%

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

Alternative 22: 62.7% accurate, 1.0× speedup

Math

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

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)))
      (exp
       (*
        (log
         (-
          (pow (cos (* (- phi1 phi2) 0.5)) 2.0)
          (*
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
           (* (cos phi2) (cos phi1)))))
        0.5)))))))

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))), exp((log((pow(cos(((phi1 - phi2) * 0.5)), 2.0) - ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1))))) * 0.5))));
}

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))), exp((log(((cos(((phi1 - phi2) * 0.5d0)) ** 2.0d0) - ((0.5d0 - (0.5d0 * cos((2.0d0 * (0.5d0 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1))))) * 0.5d0))))
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.exp((Math.log((Math.pow(Math.cos(((phi1 - phi2) * 0.5)), 2.0) - ((0.5 - (0.5 * Math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (Math.cos(phi2) * Math.cos(phi1))))) * 0.5))));
}

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.exp((math.log((math.pow(math.cos(((phi1 - phi2) * 0.5)), 2.0) - ((0.5 - (0.5 * math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (math.cos(phi2) * math.cos(phi1))))) * 0.5))))

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))), exp(Float64(log(Float64((cos(Float64(Float64(phi1 - phi2) * 0.5)) ^ 2.0) - Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))) * Float64(cos(phi2) * cos(phi1))))) * 0.5)))))
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))), exp((log(((cos(((phi1 - phi2) * 0.5)) ^ 2.0) - ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1))))) * 0.5))));
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[Exp[N[(N[Log[N[(N[Power[N[Cos[N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $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] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.7%

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

3. Applied rewrites 62.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-cos-a-rev N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-cos.f64 62.8

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 62.8

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

5. Applied rewrites 62.8%

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

Alternative 23: 61.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0}}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_1 - \lambda_2\right)\right), 0.5, -0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right) - \left(0.5 - \cos \left(-1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\right)\right) \cdot 0.5}}\right) \end{array} \]

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)))
      (exp
       (*
        (log1p
         (-
          (*
           (fma (cos (* -1.0 (- lambda1 lambda2))) 0.5 -0.5)
           (* (cos phi2) (cos phi1)))
          (- 0.5 (* (cos (* -1.0 (- phi1 phi2))) 0.5))))
        0.5)))))))

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))), exp((log1p(((fma(cos((-1.0 * (lambda1 - lambda2))), 0.5, -0.5) * (cos(phi2) * cos(phi1))) - (0.5 - (cos((-1.0 * (phi1 - phi2))) * 0.5)))) * 0.5))));
}

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))), exp(Float64(log1p(Float64(Float64(fma(cos(Float64(-1.0 * Float64(lambda1 - lambda2))), 0.5, -0.5) * Float64(cos(phi2) * cos(phi1))) - Float64(0.5 - Float64(cos(Float64(-1.0 * Float64(phi1 - phi2))) * 0.5)))) * 0.5)))))
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[Exp[N[(N[Log[1 + N[(N[(N[(N[Cos[N[(-1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 - N[(N[Cos[N[(-1.0 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.7%

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

3. Applied rewrites 62.8%

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

5. Applied rewrites 62.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_1 - \lambda_2\right)\right), 0.5, -0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right) - \left(0.5 - \cos \left(-1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\right)\right)} \cdot 0.5}}\right) \]
6. Add Preprocessing

Alternative 24: 61.3% 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 62.7%

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

3. Applied rewrites 62.8%

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

Alternative 25: 61.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (* (cos phi2) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
          (pow (sin (* 0.5 (- phi1 phi2))) 2.0))))
   (* 2.0 (* R (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(sin((0.5 * (lambda1 - lambda2))), 2.0)), pow(sin((0.5 * (phi1 - phi2))), 2.0));
	return 2.0 * (R * 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) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))
	return Float64(2.0 * Float64(R * 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[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[(R * 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 62.7%

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

2. Taylor expanded in R around 0

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

   1. lower-*.f64 N/A

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

4. Applied rewrites 62.7%

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

Alternative 26: 61.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-1 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_1 := \left(0.5 - \cos \left(-1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5\right) \cdot \cos \phi_2\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t\_1\\ \mathbf{if}\;\phi_2 \leq -0.018:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_1 + \left(0.5 - t\_0 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(t\_0, 0.5, 0.5\right) - t\_1}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 48000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* -1.0 (- phi1 phi2))))
        (t_1 (* (- 0.5 (* (cos (* -1.0 (- lambda1 lambda2))) 0.5)) (cos phi2)))
        (t_2 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) t_1)))
   (if (<= phi2 -0.018)
     (*
      (*
       (atan2
        (sqrt (+ t_1 (- 0.5 (* t_0 0.5))))
        (sqrt (- (fma t_0 0.5 0.5) t_1)))
       2.0)
      R)
     (if (<= phi2 48000.0)
       (*
        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
             (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
             (cos phi1)
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))))))
       (* 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 = cos((-1.0 * (phi1 - phi2)));
	double t_1 = (0.5 - (cos((-1.0 * (lambda1 - lambda2))) * 0.5)) * cos(phi2);
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + t_1;
	double tmp;
	if (phi2 <= -0.018) {
		tmp = (atan2(sqrt((t_1 + (0.5 - (t_0 * 0.5)))), sqrt((fma(t_0, 0.5, 0.5) - t_1))) * 2.0) * R;
	} else if (phi2 <= 48000.0) {
		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((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(-1.0 * Float64(phi1 - phi2)))
	t_1 = Float64(Float64(0.5 - Float64(cos(Float64(-1.0 * Float64(lambda1 - lambda2))) * 0.5)) * cos(phi2))
	t_2 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + t_1)
	tmp = 0.0
	if (phi2 <= -0.018)
		tmp = Float64(Float64(atan(sqrt(Float64(t_1 + Float64(0.5 - Float64(t_0 * 0.5)))), sqrt(Float64(fma(t_0, 0.5, 0.5) - t_1))) * 2.0) * R);
	elseif (phi2 <= 48000.0)
		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(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(-1.0 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 - N[(N[Cos[N[(-1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[phi2, -0.018], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$1 + N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * 0.5 + 0.5), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 48000.0], 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[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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], 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. Split input into 3 regimes

2. if phi2 < -0.0179999999999999986

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. 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 63.3%

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

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   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 61.1%

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

   17. Applied rewrites 46.3%

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

   if -0.0179999999999999986 < phi2 < 48000

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 47.5

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

   9. Applied rewrites 47.5%

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

   if 48000 < phi2

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. 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 63.3%

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

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   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 61.1%

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

   17. Applied rewrites 49.0%

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

   19. Applied rewrites 48.7%

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

Alternative 27: 61.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-1 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_1 := \left(0.5 - \cos \left(-1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5\right) \cdot \cos \phi_2\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{t\_1 + \left(0.5 - t\_0 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(t\_0, 0.5, 0.5\right) - t\_1}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -0.018:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 48000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* -1.0 (- phi1 phi2))))
        (t_1 (* (- 0.5 (* (cos (* -1.0 (- lambda1 lambda2))) 0.5)) (cos phi2)))
        (t_2
         (*
          (*
           (atan2
            (sqrt (+ t_1 (- 0.5 (* t_0 0.5))))
            (sqrt (- (fma t_0 0.5 0.5) t_1)))
           2.0)
          R)))
   (if (<= phi2 -0.018)
     t_2
     (if (<= phi2 48000.0)
       (*
        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
             (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
             (cos phi1)
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((-1.0 * (phi1 - phi2)));
	double t_1 = (0.5 - (cos((-1.0 * (lambda1 - lambda2))) * 0.5)) * cos(phi2);
	double t_2 = (atan2(sqrt((t_1 + (0.5 - (t_0 * 0.5)))), sqrt((fma(t_0, 0.5, 0.5) - t_1))) * 2.0) * R;
	double tmp;
	if (phi2 <= -0.018) {
		tmp = t_2;
	} else if (phi2 <= 48000.0) {
		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((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(-1.0 * Float64(phi1 - phi2)))
	t_1 = Float64(Float64(0.5 - Float64(cos(Float64(-1.0 * Float64(lambda1 - lambda2))) * 0.5)) * cos(phi2))
	t_2 = Float64(Float64(atan(sqrt(Float64(t_1 + Float64(0.5 - Float64(t_0 * 0.5)))), sqrt(Float64(fma(t_0, 0.5, 0.5) - t_1))) * 2.0) * R)
	tmp = 0.0
	if (phi2 <= -0.018)
		tmp = t_2;
	elseif (phi2 <= 48000.0)
		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(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(-1.0 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 - N[(N[Cos[N[(-1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[N[(t$95$1 + N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * 0.5 + 0.5), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -0.018], t$95$2, If[LessEqual[phi2, 48000.0], 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[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.0179999999999999986 or 48000 < phi2

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. 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 63.3%

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

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   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 61.1%

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

   17. Applied rewrites 46.3%

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

   if -0.0179999999999999986 < phi2 < 48000

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 47.5

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

   9. Applied rewrites 47.5%

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

Alternative 28: 58.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\\ t_1 := \cos \left(-1 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_2 := \cos \left(-1 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_3 := t\_0 - \mathsf{fma}\left(t\_2, 0.5, -0.5\right) \cdot \cos \phi_1\\ t_4 := 0.5 - t\_2 \cdot 0.5\\ t_5 := t\_4 \cdot \cos \phi_2\\ t_6 := \mathsf{fma}\left(t\_4, \cos \phi_1, t\_0\right)\\ \mathbf{if}\;\phi_1 \leq -1.7 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{elif}\;\phi_1 \leq 2.85 \cdot 10^{-5}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_5 + \left(0.5 - t\_1 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(t\_1, 0.5, 0.5\right) - t\_5}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_6}}{\sqrt{1 - t\_6}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1))))))
        (t_1 (cos (* -1.0 (- phi1 phi2))))
        (t_2 (cos (* -1.0 (- lambda1 lambda2))))
        (t_3 (- t_0 (* (fma t_2 0.5 -0.5) (cos phi1))))
        (t_4 (- 0.5 (* t_2 0.5)))
        (t_5 (* t_4 (cos phi2)))
        (t_6 (fma t_4 (cos phi1) t_0)))
   (if (<= phi1 -1.7e+14)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (if (<= phi1 2.85e-5)
       (*
        (*
         (atan2
          (sqrt (+ t_5 (- 0.5 (* t_1 0.5))))
          (sqrt (- (fma t_1 0.5 0.5) t_5)))
         2.0)
        R)
       (* (* (atan2 (sqrt t_6) (sqrt (- 1.0 t_6))) 2.0) R)))))

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 * phi1))));
	double t_1 = cos((-1.0 * (phi1 - phi2)));
	double t_2 = cos((-1.0 * (lambda1 - lambda2)));
	double t_3 = t_0 - (fma(t_2, 0.5, -0.5) * cos(phi1));
	double t_4 = 0.5 - (t_2 * 0.5);
	double t_5 = t_4 * cos(phi2);
	double t_6 = fma(t_4, cos(phi1), t_0);
	double tmp;
	if (phi1 <= -1.7e+14) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else if (phi1 <= 2.85e-5) {
		tmp = (atan2(sqrt((t_5 + (0.5 - (t_1 * 0.5)))), sqrt((fma(t_1, 0.5, 0.5) - t_5))) * 2.0) * R;
	} else {
		tmp = (atan2(sqrt(t_6), sqrt((1.0 - t_6))) * 2.0) * R;
	}
	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 * phi1)))))
	t_1 = cos(Float64(-1.0 * Float64(phi1 - phi2)))
	t_2 = cos(Float64(-1.0 * Float64(lambda1 - lambda2)))
	t_3 = Float64(t_0 - Float64(fma(t_2, 0.5, -0.5) * cos(phi1)))
	t_4 = Float64(0.5 - Float64(t_2 * 0.5))
	t_5 = Float64(t_4 * cos(phi2))
	t_6 = fma(t_4, cos(phi1), t_0)
	tmp = 0.0
	if (phi1 <= -1.7e+14)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	elseif (phi1 <= 2.85e-5)
		tmp = Float64(Float64(atan(sqrt(Float64(t_5 + Float64(0.5 - Float64(t_1 * 0.5)))), sqrt(Float64(fma(t_1, 0.5, 0.5) - t_5))) * 2.0) * R);
	else
		tmp = Float64(Float64(atan(sqrt(t_6), sqrt(Float64(1.0 - t_6))) * 2.0) * R);
	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 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-1.0 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(-1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 - N[(N[(t$95$2 * 0.5 + -0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[phi1, -1.7e+14], 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, 2.85e-5], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$5 + N[(0.5 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$1 * 0.5 + 0.5), $MachinePrecision] - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[t$95$6], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$6), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.7e14

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-*.f64 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. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. mult-flip N/A

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

      5. div-flip N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-/.f64 44.5

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

   9. Applied rewrites 44.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. mult-flip N/A

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

       5. div-flip N/A

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

       6. lower-unsound-/.f64 N/A

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

       7. lower-unsound-/.f64 44.4

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

   11. Applied rewrites 44.4%

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

   13. Applied rewrites 40.5%

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

   15. Applied rewrites 43.5%

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

   if -1.7e14 < phi1 < 2.8500000000000002e-5

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. 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 63.3%

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

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   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 61.1%

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

   17. Applied rewrites 46.3%

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

   if 2.8500000000000002e-5 < phi1

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-*.f64 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. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. mult-flip N/A

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

      5. div-flip N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-/.f64 44.5

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

   9. Applied rewrites 44.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. mult-flip N/A

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

       5. div-flip N/A

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

       6. lower-unsound-/.f64 N/A

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

       7. lower-unsound-/.f64 44.4

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

   11. Applied rewrites 44.4%

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

   13. Applied rewrites 43.5%

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

Alternative 29: 58.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-1 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_1 := \cos \left(-1 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_2 := \left(0.5 - t\_1 \cdot 0.5\right) \cdot \cos \phi_2\\ t_3 := \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right) - \mathsf{fma}\left(t\_1, 0.5, -0.5\right) \cdot \cos \phi_1\\ t_4 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{if}\;\phi_1 \leq -1.7 \cdot 10^{+14}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\phi_1 \leq 2.85 \cdot 10^{-5}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_2 + \left(0.5 - t\_0 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(t\_0, 0.5, 0.5\right) - t\_2}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* -1.0 (- phi1 phi2))))
        (t_1 (cos (* -1.0 (- lambda1 lambda2))))
        (t_2 (* (- 0.5 (* t_1 0.5)) (cos phi2)))
        (t_3
         (-
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))
          (* (fma t_1 0.5 -0.5) (cos phi1))))
        (t_4 (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))))
   (if (<= phi1 -1.7e+14)
     t_4
     (if (<= phi1 2.85e-5)
       (*
        (*
         (atan2
          (sqrt (+ t_2 (- 0.5 (* t_0 0.5))))
          (sqrt (- (fma t_0 0.5 0.5) t_2)))
         2.0)
        R)
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((-1.0 * (phi1 - phi2)));
	double t_1 = cos((-1.0 * (lambda1 - lambda2)));
	double t_2 = (0.5 - (t_1 * 0.5)) * cos(phi2);
	double t_3 = (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))) - (fma(t_1, 0.5, -0.5) * cos(phi1));
	double t_4 = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	double tmp;
	if (phi1 <= -1.7e+14) {
		tmp = t_4;
	} else if (phi1 <= 2.85e-5) {
		tmp = (atan2(sqrt((t_2 + (0.5 - (t_0 * 0.5)))), sqrt((fma(t_0, 0.5, 0.5) - t_2))) * 2.0) * R;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(-1.0 * Float64(phi1 - phi2)))
	t_1 = cos(Float64(-1.0 * Float64(lambda1 - lambda2)))
	t_2 = Float64(Float64(0.5 - Float64(t_1 * 0.5)) * cos(phi2))
	t_3 = Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))) - Float64(fma(t_1, 0.5, -0.5) * cos(phi1)))
	t_4 = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))))
	tmp = 0.0
	if (phi1 <= -1.7e+14)
		tmp = t_4;
	elseif (phi1 <= 2.85e-5)
		tmp = Float64(Float64(atan(sqrt(Float64(t_2 + Float64(0.5 - Float64(t_0 * 0.5)))), sqrt(Float64(fma(t_0, 0.5, 0.5) - t_2))) * 2.0) * R);
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(-1.0 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * 0.5 + -0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$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]}, If[LessEqual[phi1, -1.7e+14], t$95$4, If[LessEqual[phi1, 2.85e-5], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$2 + N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * 0.5 + 0.5), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.7e14 or 2.8500000000000002e-5 < phi1

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-*.f64 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. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. mult-flip N/A

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

      5. div-flip N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. lower-unsound-/.f64 44.5

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

   9. Applied rewrites 44.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. mult-flip N/A

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

       5. div-flip N/A

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

       6. lower-unsound-/.f64 N/A

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

       7. lower-unsound-/.f64 44.4

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

   11. Applied rewrites 44.4%

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

   13. Applied rewrites 40.5%

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

   15. Applied rewrites 43.5%

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

   if -1.7e14 < phi1 < 2.8500000000000002e-5

   1. Initial program 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. 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 63.3%

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

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   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 61.1%

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

   17. Applied rewrites 46.3%

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

Alternative 30: 49.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-1 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ t_3 := \left(0.5 - \cos \left(-1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5\right) \cdot \cos \phi_2\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_3 + \left(0.5 - t\_0 \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(t\_0, 0.5, 0.5\right) - t\_3}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* -1.0 (- phi1 phi2))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_1) t_1)))
        (t_3
         (* (- 0.5 (* (cos (* -1.0 (- lambda1 lambda2))) 0.5)) (cos phi2))))
   (if (<= (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))) 2e-17)
     (*
      (*
       (atan2
        (sqrt
         (fma
          (cos phi2)
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (pow (sin (* -0.5 phi2)) 2.0)))
        (sqrt
         (-
          1.0
          (fma
           (* phi1 0.5)
           (* phi1 0.5)
           (*
            (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
            (cos phi1))))))
       2.0)
      R)
     (*
      (*
       (atan2
        (sqrt (+ t_3 (- 0.5 (* t_0 0.5))))
        (sqrt (- (fma t_0 0.5 0.5) t_3)))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((-1.0 * (phi1 - phi2)));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1);
	double t_3 = (0.5 - (cos((-1.0 * (lambda1 - lambda2))) * 0.5)) * cos(phi2);
	double tmp;
	if ((2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2)))) <= 2e-17) {
		tmp = (atan2(sqrt(fma(cos(phi2), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((-0.5 * phi2)), 2.0))), sqrt((1.0 - fma((phi1 * 0.5), (phi1 * 0.5), ((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))) * cos(phi1)))))) * 2.0) * R;
	} else {
		tmp = (atan2(sqrt((t_3 + (0.5 - (t_0 * 0.5)))), sqrt((fma(t_0, 0.5, 0.5) - t_3))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(-1.0 * Float64(phi1 - phi2)))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))
	t_3 = Float64(Float64(0.5 - Float64(cos(Float64(-1.0 * Float64(lambda1 - lambda2))) * 0.5)) * cos(phi2))
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))) <= 2e-17)
		tmp = Float64(Float64(atan(sqrt(fma(cos(phi2), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(-0.5 * phi2)) ^ 2.0))), sqrt(Float64(1.0 - fma(Float64(phi1 * 0.5), Float64(phi1 * 0.5), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))) * cos(phi1)))))) * 2.0) * R);
	else
		tmp = Float64(Float64(atan(sqrt(Float64(t_3 + Float64(0.5 - Float64(t_0 * 0.5)))), sqrt(Float64(fma(t_0, 0.5, 0.5) - t_3))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(-1.0 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $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[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 - N[(N[Cos[N[(-1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-17], N[(N[(N[ArcTan[N[Sqrt[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] / N[Sqrt[N[(1.0 - N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$3 + N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$0 * 0.5 + 0.5), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $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))))))))) < 2.00000000000000014e-17

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in phi1 around 0

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

       1. lower-*.f64 22.4

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

   13. Applied rewrites 22.4%

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

   15. Applied rewrites 19.7%

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

   16. Taylor expanded in phi1 around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-cos.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-pow.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-*.f64 23.4

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

   18. Applied rewrites 23.4%

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

   if 2.00000000000000014e-17 < (*.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 62.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 61.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 77.5%

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

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. 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 63.3%

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

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   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 61.1%

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

   17. Applied rewrites 46.3%

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

Alternative 31: 33.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)\\ t_2 := \sqrt{t\_1}\\ t_3 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ \mathbf{if}\;\phi_1 \leq -1.15:\\ \;\;\;\;\left(\tan^{-1}_* \frac{t\_2}{\sqrt{1 - t\_3}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 1.3 \cdot 10^{-30}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_1}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{t\_2}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1
         (fma
          (* phi1 0.5)
          (* phi1 0.5)
          (*
           (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
           (cos phi1))))
        (t_2 (sqrt t_1))
        (t_3 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0))))
   (if (<= phi1 -1.15)
     (* (* (atan2 t_2 (sqrt (- 1.0 t_3))) 2.0) R)
     (if (<= phi1 1.3e-30)
       (* (* (atan2 (sqrt t_3) (sqrt (- 1.0 t_1))) 2.0) R)
       (*
        (*
         (atan2
          t_2
          (sqrt (- 1.0 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))))
         2.0)
        R)))))

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((phi1 * 0.5), (phi1 * 0.5), ((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))) * cos(phi1)));
	double t_2 = sqrt(t_1);
	double t_3 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double tmp;
	if (phi1 <= -1.15) {
		tmp = (atan2(t_2, sqrt((1.0 - t_3))) * 2.0) * R;
	} else if (phi1 <= 1.3e-30) {
		tmp = (atan2(sqrt(t_3), sqrt((1.0 - t_1))) * 2.0) * R;
	} else {
		tmp = (atan2(t_2, sqrt((1.0 - fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0))))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = fma(Float64(phi1 * 0.5), Float64(phi1 * 0.5), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))) * cos(phi1)))
	t_2 = sqrt(t_1)
	t_3 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	tmp = 0.0
	if (phi1 <= -1.15)
		tmp = Float64(Float64(atan(t_2, sqrt(Float64(1.0 - t_3))) * 2.0) * R);
	elseif (phi1 <= 1.3e-30)
		tmp = Float64(Float64(atan(sqrt(t_3), sqrt(Float64(1.0 - t_1))) * 2.0) * R);
	else
		tmp = Float64(Float64(atan(t_2, sqrt(Float64(1.0 - fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))))) * 2.0) * R);
	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[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.15], N[(N[(N[ArcTan[t$95$2 / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 1.3e-30], N[(N[(N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], N[(N[(N[ArcTan[t$95$2 / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.1499999999999999

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in phi1 around 0

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

       1. lower-*.f64 22.4

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

   13. Applied rewrites 22.4%

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

   15. Applied rewrites 19.7%

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

   16. Taylor expanded in phi1 around 0

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

       1. lower--.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-cos.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-pow.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-*.f64 29.6

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

   18. Applied rewrites 29.6%

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

   if -1.1499999999999999 < phi1 < 1.29999999999999993e-30

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in phi1 around 0

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

       1. lower-*.f64 22.4

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

   13. Applied rewrites 22.4%

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

   15. Applied rewrites 19.7%

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

   16. Taylor expanded in phi1 around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-cos.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-pow.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-*.f64 23.4

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

   18. Applied rewrites 23.4%

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

   if 1.29999999999999993e-30 < phi1

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in phi1 around 0

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

       1. lower-*.f64 22.4

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

   13. Applied rewrites 22.4%

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

   15. Applied rewrites 19.7%

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

   16. Taylor expanded in phi2 around 0

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

       1. lower--.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-cos.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-pow.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-*.f64 29.4

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

   18. Applied rewrites 29.4%

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

Alternative 32: 33.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -1.46:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 1.3 \cdot 10^{-30}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, t\_1, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - t\_0}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (* phi1 0.5)
          (* phi1 0.5)
          (*
           (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
           (cos phi1))))
        (t_1 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_2
         (*
          (*
           (atan2
            (sqrt t_0)
            (sqrt (- 1.0 (fma (cos phi1) t_1 (pow (sin (* 0.5 phi1)) 2.0)))))
           2.0)
          R)))
   (if (<= phi1 -1.46)
     t_2
     (if (<= phi1 1.3e-30)
       (*
        (*
         (atan2
          (sqrt (fma (cos phi2) t_1 (pow (sin (* -0.5 phi2)) 2.0)))
          (sqrt (- 1.0 t_0)))
         2.0)
        R)
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((phi1 * 0.5), (phi1 * 0.5), ((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))) * cos(phi1)));
	double t_1 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_2 = (atan2(sqrt(t_0), sqrt((1.0 - fma(cos(phi1), t_1, pow(sin((0.5 * phi1)), 2.0))))) * 2.0) * R;
	double tmp;
	if (phi1 <= -1.46) {
		tmp = t_2;
	} else if (phi1 <= 1.3e-30) {
		tmp = (atan2(sqrt(fma(cos(phi2), t_1, pow(sin((-0.5 * phi2)), 2.0))), sqrt((1.0 - t_0))) * 2.0) * R;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(phi1 * 0.5), Float64(phi1 * 0.5), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))) * cos(phi1)))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_2 = Float64(Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - fma(cos(phi1), t_1, (sin(Float64(0.5 * phi1)) ^ 2.0))))) * 2.0) * R)
	tmp = 0.0
	if (phi1 <= -1.46)
		tmp = t_2;
	elseif (phi1 <= 1.3e-30)
		tmp = Float64(Float64(atan(sqrt(fma(cos(phi2), t_1, (sin(Float64(-0.5 * phi2)) ^ 2.0))), sqrt(Float64(1.0 - t_0))) * 2.0) * R);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * t$95$1 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -1.46], t$95$2, If[LessEqual[phi1, 1.3e-30], N[(N[(N[ArcTan[N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * t$95$1 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.46 or 1.29999999999999993e-30 < phi1

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in phi1 around 0

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

       1. lower-*.f64 22.4

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

   13. Applied rewrites 22.4%

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

   15. Applied rewrites 19.7%

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

   16. Taylor expanded in phi2 around 0

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

       1. lower--.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-cos.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-pow.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-*.f64 29.4

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

   18. Applied rewrites 29.4%

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

   if -1.46 < phi1 < 1.29999999999999993e-30

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in phi1 around 0

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

       1. lower-*.f64 22.4

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

   13. Applied rewrites 22.4%

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

   15. Applied rewrites 19.7%

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

   16. Taylor expanded in phi1 around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-cos.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-pow.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-*.f64 23.4

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

   18. Applied rewrites 23.4%

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

Alternative 33: 27.7% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  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
       (* phi1 0.5)
       (* phi1 0.5)
       (*
        (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
        (cos phi1))))
     0.5)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return 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((phi1 * 0.5), (phi1 * 0.5), ((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))) * cos(phi1)))), 0.5)));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return 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(phi1 * 0.5), Float64(phi1 * 0.5), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))) * cos(phi1)))) ^ 0.5))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := 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[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.7%

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

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 47.3

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

4. Applied rewrites 47.3%

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

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
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 47.5

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

7. Applied rewrites 47.5%

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

8. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 32.0

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

10. Applied rewrites 32.0%

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

11. Taylor expanded in phi1 around 0

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

    1. lower-*.f64 22.4

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

13. Applied rewrites 22.4%

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

15. Applied rewrites 27.7%

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

Alternative 34: 27.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}} \leq 0.166:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(-1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(0.5 \cdot \phi_1\right) \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}^{0.5}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (+ 1.0 (* -0.5 (pow phi1 2.0)))
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (pow (* 0.5 phi1) 2.0)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_1) t_1))))
   (if (<= (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))) 0.166)
     (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
     (*
      (*
       (atan2
        (sqrt
         (fma
          (* phi1 0.5)
          (* phi1 0.5)
          (*
           (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
           (cos phi1))))
        (pow
         (-
          1.0
          (fma
           (- 0.5 (* (cos (* -1.0 (- lambda1 lambda2))) 0.5))
           (cos phi1)
           (* (* 0.5 phi1) (* 0.5 phi1))))
         0.5))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((1.0 + (-0.5 * pow(phi1, 2.0))), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow((0.5 * phi1), 2.0));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1);
	double tmp;
	if ((2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2)))) <= 0.166) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = (atan2(sqrt(fma((phi1 * 0.5), (phi1 * 0.5), ((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))) * cos(phi1)))), pow((1.0 - fma((0.5 - (cos((-1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), ((0.5 * phi1) * (0.5 * phi1)))), 0.5)) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (Float64(0.5 * phi1) ^ 2.0))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))) <= 0.166)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(Float64(phi1 * 0.5), Float64(phi1 * 0.5), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))) * cos(phi1)))), (Float64(1.0 - fma(Float64(0.5 - Float64(cos(Float64(-1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), Float64(Float64(0.5 * phi1) * Float64(0.5 * phi1)))) ^ 0.5)) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $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]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $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[(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$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.166], 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], N[(N[(N[ArcTan[N[Sqrt[N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $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[(0.5 * phi1), $MachinePrecision] * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $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))))))))) < 0.166000000000000009

   1. Initial program 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in phi1 around 0

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

       1. lower-*.f64 22.4

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

   13. Applied rewrites 22.4%

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

   14. Taylor expanded in phi1 around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 22.4

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

   16. Applied rewrites 22.4%

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

   17. Taylor expanded in phi1 around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 22.4

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

   19. Applied rewrites 22.4%

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

   if 0.166000000000000009 < (*.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 62.7%

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

   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 47.3

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

   4. Applied rewrites 47.3%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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 47.5

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

   7. Applied rewrites 47.5%

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

   8. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   11. Taylor expanded in phi1 around 0

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

       1. lower-*.f64 22.4

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

   13. Applied rewrites 22.4%

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

   15. Applied rewrites 19.7%

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

   17. Applied rewrites 25.0%

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

Alternative 35: 25.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[(N[ArcTan[N[Sqrt[N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $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[(0.5 * phi1), $MachinePrecision] * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 62.7%

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

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 47.3

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

4. Applied rewrites 47.3%

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

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
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 47.5

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

7. Applied rewrites 47.5%

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

8. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 32.0

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

10. Applied rewrites 32.0%

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

11. Taylor expanded in phi1 around 0

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

    1. lower-*.f64 22.4

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

13. Applied rewrites 22.4%

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

15. Applied rewrites 19.7%

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

17. Applied rewrites 25.0%

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

Alternative 36: 19.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

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 47.3

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

4. Applied rewrites 47.3%

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

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
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 47.5

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

7. Applied rewrites 47.5%

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

8. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 32.0

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

10. Applied rewrites 32.0%

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

11. Taylor expanded in phi1 around 0

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

    1. lower-*.f64 22.4

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

13. Applied rewrites 22.4%

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

15. Applied rewrites 19.7%

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

16. Taylor expanded in phi1 around 0

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

    1. lower-+.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-pow.f64 19.7

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

18. Applied rewrites 19.7%

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

19. Taylor expanded in phi1 around 0

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

    1. lower-+.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-pow.f64 19.7

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

21. Applied rewrites 19.7%

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

Reproduce

herbie shell --seed 2025173 
(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))))))))))